Apple Pie Cookies
['1/4 cup granulated sugar', '1 teaspoon kosher salt', '2 1/2 cups all-purpose flour, plus more for surface', '1 cup (2 sticks) unsalted butter, cut into small pieces, frozen 15 minutes', '1 tablespoon chilled apple cider vinegar', '2 medium Honeycrisp, Braeburn, and/or Golden Delicious apples (about 1 pound total)', '1/4 cup light brown sugar', '2 teaspoons fresh lemon juice', '1/2 teaspoon ground cinnamon', '1/4 teaspoon kosher salt', '2 teaspoons all-purpose flour, plus more for surface', '1 large egg', '1 tablespoon milk', 'Demerara or raw sugar (for sprinkling; optional)', 'A 3" round cookie or biscuit cutter; a small (about 1 1/2") leaf-shaped or round cookie cutter']
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.
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.
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.
Preheat oven to 375°F. Line a rimmed baking sheet with parchment paper. Beat egg and milk in a small bowl.
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.
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.
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.
Brush top of sandwiches with egg wash and sprinkle with demerara sugar, if using.
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.
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.
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