Fried apple pie, more New Orleans treats

To make the dough
Sift the flour, baking powder and salt into the bowl of a stand mixer (use a large bowl if mixing by hand). Add the melted shortening, eggs and milk. Using the paddle attachment, mix the dough on low speed until the dry ingredients are moistened. Increase the speed to medium and knead the dough until it is no longer sticky, about 2 to 3 minutes, adding more flour, 1 tablespoon at a time, if the dough seems very wet or sticky. (If mixing by hand, it will take about 5 minutes to knead the dough.) Divide the dough in half, wrap each half in plastic wrap, and refrigerate for 1 hour (because the dough turns gray, it is best used within 6 hours of making; see "Make Ahead" on the facing page for freezing instructions).

To make the filling
Meanwhile, make the apple filling. Line a rimmed baking sheet with parchment paper. Whisk 1/4 cup of the apple juice or cider with the cornstarch in a small bowl and set aside. Place the apples, sugar, nutmeg and cinnamon in a large bowl. Wrap one-half of the vanilla bean well in plastic wrap, and reserve for another use. Slice the remaining half lengthwise down the middle. Open the pod and use the knife to scrape out the seeds; discard the pod. Add the seeds to the apples and toss with the sugar and spices.

Melt the butter in a large pot over medium-high heat. Once melted, let the butter simmer for 20 seconds, then stir in the apples. Cook with a lid askew until the apples start to soften, about 5 to 7 minutes, stirring occasionally. Add the remaining 1/4 cup of apple juice or cider and simmer, stirring occasionally, until the liquid is reduced by half, about 2 to 3 minutes. Stir in the cornstarch slurry and cook until the mixture is thickened, 3 to 4 minutes. Transfer to the prepared baking sheet (or a large bowl) and cool to room temperature, and then cover with plastic wrap and refrigerate until cold.

To make the piecrusts
Once the apples have chilled, make the piecrusts. Line a rimmed baking sheet with parchment paper. Place a small dish of water next to your work surface. Generously flour the work surface and roll one dough ball into a 1/8-inch-thick circle. Use a 3 1/2- to 4-inch biscuit or cookie cutter to cut out rounds from the dough. Place the circles on the prepared baking sheet, cover with a sheet of parchment paper and place them in the refrigerator to keep cool. Set the dough scraps aside and reflour your work surface. Repeat with the second piece of dough and refrigerate the stamped-out rounds. Press all of the scraps together, roll them into a 1/8-inch-thick circle, cut out rounds, and refrigerate. Discard the leftover scraps. (Discard the scraps after rolling the second time; the dough gets too tough to roll a third time.)

To assemble the pies
Place 1 1/2 to 2 tablespoons of cold apple filling in the center of each dough circle. Using your finger or a pastry brush, moisten the edge of the bottom half of the circle with water and fold the top half over, bringing the edges together and pressing them tightly to seal. Using firm pressure, crimp the edges of the dough using an upturned fork. Refrigerate the filled pies while you heat the oil. (The pies can be made up to 6 hours before frying.)

Line a plate with paper towels. Pour enough peanut oil into a large pot to fill it to a 3-inch depth and bring to a temperature of 375°F over high heat. Fry a few turnovers (you don't want to fry too many at one time, otherwise the temperature of the oil will drop and the turnovers will become greasy) until all sides are golden brown, about 3 to 5 minutes, turning them over often. Transfer to the prepared plate to drain while you fry the remaining turnovers. Serve warm or at room temperature with plenty of confectioners' sugar on top.

Make ahead
The apple filling can be made up to two days ahead, and the stamped-out dough circles can be frozen for up to six months. Freeze the dough circles flat on the parchment paper-lined baking sheet (if you need to stack the dough, separate the layers with parchment paper). Once frozen solid, transfer the circles to a resealable freezer bag. To use, return the circles to a parchment paper-lined baking sheet, cover them with another sheet, defrost in your refrigerator overnight, and then fill. The filled and crimped pies can be refrigerated for up to six hours before frying.

To make the cake
Heat the oven to 350 degrees F. Grease two 9-inch cake pans with 1 tablespoon of butter each. Add 2 tablespoons of the flour to each pan and shake the pans to coat the bottom and sides. Tap out the excess flour and set the pans aside.

Sift the remaining 3 cups of flour with the cocoa, baking powder, baking soda, and salt, and set aside.

In the bowl of a stand mixer (or in a large bowl if using a hand mixer) cream the remaining butter with the brown sugar, food coloring, and vanilla on low to combine. Increase the mixer speed to medium-high and beat until aerated and pale, about 2 minutes. Reduce the speed to medium and add the eggs, one at a time, beating thoroughly between each addition and using a rubber spatula to scrape the sides and bottom of the bowl as necessary. Reduce the speed to low and add one-third of the dry ingredients followed by half of the buttermilk. Repeat, finishing with the final third of the dry mix. Scrape down the bottom and sides of the bowl and divide the batter between the two prepared cake pans, spreading it out as evenly as possible.

Bake until a cake tester inserted into the center comes out clean and the center of the cake resists slight pressure, about 40 minutes. Cool on a wire rack for 15 minutes, and then run a paring knife around the edges of each pan to release the cake from the sides; invert the cakes onto the cooling rack. Cool for 1 hour, and then wrap each cake in plastic wrap for at least a few hours.

To make the frosting
Beat the cream cheese, butter, and vanilla together in the bowl of a stand mixer (or in a large bowl if using a hand mixer) on low speed to combine. Increase the speed to medium-high and beat until aerated and light, about 2 minutes. Stop the mixer and add a few cups of the confectioners' sugar, incorporating it into the cream cheese mixture on low speed until combined. Repeat with the remaining sugar, adding it to the mixer in two additions. Once all of the sugar is added, increase the speed to medium-high and beat until fluffy, about 1 minute.

To assemble the cake
Unwrap the cake layers. Slice off the rounded top 1/8 inch of each cake and place the trimmed-away portion in the bowl of a food processor. Slice each cake in half horizontally (you'll end up with 4 layers), working over a baking sheet to catch any crumb. Add the crumbs to the food processor and pulse until fine.

Place one cake layer on a cake round or large plate (make sure that the diameter of the plate is at least 1 inch larger than the cake). Use an offset spatula to evenly spread a heaping 3/4 cup of frosting on the first cake layer. Repeat with the remaining three cake layers, ending with a bottom half of a cake on top, browned-side up (so you don't get cake crumbs in the frosting). Spread the remaining frosting over the top and sides of the cake (the sides don't have to look perfect — you're going to cover them with cake crumbs anyway). Gently press a handful of the reserved crumbs into the side of the cake until all of the sides are evenly coated. Refrigerate for at least 2 hours before serving.

Make ahead
You can make the cake layers up to three days in advance. The filled and frosted cake keeps for up to three days in the refrigerator. Let it stand at room temperature for at least 20 minutes before slicing.

Variation: Red velvet cupcakes
Follow the recipe above, making a half-batch of the batter (unless you have two 12-cup muffin tins, in which case you can make a full-size batch) and decreasing the baking powder to 1 teaspoon. Bake until the cupcakes are domes and resist slight pressure, 12 to 15 minutes. Cool completely before frosting with a half batch of the cream cheese frosting.

Heat the oven to 375 degrees F. Line a baking sheet with parchment paper. Place the puff pastry sheet on your work surface and cut out a 10-inch circle. Set the circle onto the prepared baking sheet. Prick the pastry all over with a fork and refrigerate.

Place 3/4 cup of sugar in a small saucepan and cover with 1/4 cup of water. Gently stir with a spoon to make sure all of the sugar is wet (it should have the consistency of wet sand), place a cover on slightly askew, and bring to a boil over medium-high heat. Keep the mixture covered until the syrup is clear and producing syrupy-looking medium-size bubbles, 3 to 4 minutes. Remove the cover and continue to cook until the sugar is a light butterscotch color and its temperature reaches 320 degrees F. Turn off the heat (the sugar will continue to cook in the pan even though the heat is off). Once the temperature reaches 350°F (this will take only a few minutes), whisk in the butter, 1/2 tablespoon (1 piece) at a time, waiting until each addition is completely incorporated before adding the next. Stir in the vanilla and the salt, and pour the caramel into a 10-inch cast iron skillet.

Cover the caramel with the sliced potatoes, starting in the center and overlapping in a spiraling outward circle as you go. Top with the puff pastry circle. Beat the egg and the milk together and brush over the pastry, and then sprinkle with the remaining 1 tablespoon of sugar. Bake until the edges are deep amber and the pastry is puffed and golden, 40 to 45 minutes. Remove from the oven and cool for 10 minutes before inverting onto a large plate (make sure the diameter of the plate is larger than 10 inches!). Slice into wedges and serve with or without ice cream.