Step 1

PreparationMake the pie dough: Pulse sugar, salt, and 2 1/2 cups flour in a food processor to combine. Add butter and pulse until coarse, pea-sized crumbs form, about 10 seconds. With the motor running, add vinegar and 1/4 cup ice water and process until dough just comes together, about 30 seconds. Squeeze a small amount of dough between your fingertips; if it's very crumbly, add more ice water, 1 Tbsp. at a time (2 Tbsp. maximum). Do not overprocess.

Step 2

Turn out dough onto a lightly floured surface and form into a rough ball. Knead a few times to combine, then divide into 2 equal balls. Flatten each ball into a disc with smooth edges and no cracks. Cover with plastic wrap and chill at least 1 hour or up to overnight.

Step 3

Make the filling: Peel, core, and slice apples into 1/4"-thick slices. Coarsely chop slices. Transfer to a large pot and add brown sugar, lemon juice, cinnamon, salt, 2 tsp. flour, and 2 Tbsp. water; stir to combine. Cook over medium heat, stirring occasionally and adding water 1 Tbsp. at a time if bottom starts to brown, until apples are softened but still have a little bite, about 10 minutes. Let cool to room temperature.

Step 4

Bake the cookies: Preheat oven to 375°F. Line a rimmed baking sheet with parchment paper. Beat egg and milk in a small bowl.

Step 5

Roll 1 disc of dough to an 1/8"-thick round about 13" in diameter. Using 3" cutter, cut dough into 12 rounds. Transfer rounds to prepared baking sheet. Spoon 1 heaping Tbsp. filling onto each round (do not overfill). Chill on sheet until ready to use.

Step 6

Roll second disc of dough to an 1/8"-thick round about 13" in diameter. Using 3" cutter, cut into 12 more rounds. Using leaf-shaped cutter, cut a hole in center of each round.

Step 7

Brush edges of rounds with apple filling with egg wash. Place cut-out rounds over filled rounds to create sandwiches, then press down lightly around edges to seal completely. Gently press down in center to even out filling.

Step 8

Brush top of sandwiches with egg wash and sprinkle with demerara sugar, if using.

Step 9

Bake cookies until golden brown and crisp on the bottom, 30–35 minutes. Let cool on baking sheet 5 minutes, then transfer to a wire rack and let cool completely.

Step 10

Do Ahead Cookies can be baked 2 days ahead. Loosely cover with foil and store at room temperature. Do not store in a sealed container or they will become too soft.

Step 11

Cooks' NoteThe cut-out centers of the cookies can be baked, too. Preheat oven to 375°F. Arrange cookies on a baking sheet and bake until golden brown and crisp, 12-15 minutes.