Make the dough: In a food processor, pulse the flours, sugar, salt, and cayenne to combine. Add the butter and pulse until the pieces are pea-sized. Combine the vinegar and ice water. Slowly drizzle in half while pulsing the food processor until a shaggy dough forms. Pause, and test by squeezing a clump between your fingers—if it holds, the dough is ready. If it is still crumbly, drizzle a little more of the liquid while pulsing, then test again. Evenly divide the dough between two pieces of plastic wrap. Use the wrap to shape one portion into a rectangle and tightly seal. Pat the second one into a disk and save for another use. Refrigerate the dough for at least 20 minutes or up to 3 days (you may freeze the second disk, up to 3 months).

Onto the filling: Melt the butter in a heavy 9-inch skillet over medium heat. Once the foaming subsides, cook until the butter is honey-colored and smells nutty, about 5 minutes, occasionally swirling the pan to prevent burning.

While the butter is browning, peel, halve, and core apples, then cut into ¼-inch-thick slices.

Add the apple slices to the brown butter, toss to coat, and cook for 2 minutes. Add the cinnamon, ginger, nutmeg, and salt and stir to coat. Now add the brown sugar and stir again. Add the lemon zest, followed by the juice, bitters, and cornstarch, and stir. Gently press the apples into an even layer and cook for about 5 more minutes. Once the apple slices are soft but still intact, remove the pan from the heat. Allow them to cool to room temperature. This step can be done 1 day in advance.

Roll out the dough: Roll the dough rectangle to ⅛-inch-thick, about 11 by 18 inches (I like to do this by lightly flouring a piece of parchment, then unwrapping the dough, flouring it and placing it on the parchment, then placing the plastic on top - keeps it neat and makes a cinch of transferring the dough on the parchment to chill as needed). If the dough becomes smeary or flabby at any point, transfer to a sheet pan and chill in the refrigerator for 10 to 20 minutes.

Use a 3-inch round biscuit or cookie cutter and make 12 cutouts in the dough. Arrange them on a parchment-lined sheet pan. Reroll scraps into a 9 x 6” rectangle, then cut into ¼-inch wide strips along the long edge. Arrange these on another piece of parchment set on a small tray and refrigerate.

Shingle 3 to 4 apple slices on each 3-inch circle, then replace the pan back in the fridge.

Cut the long strips into 3 inch lengths and lay them in a square frame - essentially a single, wide lattice around the edge of each cookie. Dab the ends where they overlap with a little water to affix. Holding a cookie in your hand, roll the bottom pastry edge all the way around, making a tidy border and sealing in the lattice edges. Repeat with remaining cookies.

Lightly paint each pastry surface with buttermilk, then sprinkle Demerara sugar onto each (½ teaspoon or so per cookie).

Freeze cookies for at least 30 minutes or up to 4 hours.

When you’re ready to bake, heat the oven to 400°F. Line 2 sheet pans with parchment. Place 4 cookies on each, each in their own corner. Bake for 25 minutes, until the crust is golden. After 15 minutes, rotate the pans and check the pastry—you can cover any especially dark pastry with foil strips as needed, continuing to bake until all are deeply golden.

Cool the cookies on a wire rack to room temperature and bake the remaining 4 cookies as you did the first batch. Store any leftover cookies at room temperature, wrapped in foil, for up to 2 days.