Melt the butter in a medium skillet over medium heat. Add the diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Stir and cook for 5–6 minutes until the apples soften slightly and release juices. Sprinkle in the flour and cook 1 more minute until the mixture thickens. Remove from heat and let cool for 10 minutes. The filling should look glossy and hold together when spooned.
Unroll the pie crusts on a lightly floured surface. Use a 3-inch round cutter (or the rim of a glass) to cut as many circles as possible—aim for 16 total. Re-roll scraps once if needed. Place half the rounds on a parchment-lined baking sheet.
Spoon about 1 tablespoon of cooled filling onto the center of each bottom round. Brush the edges lightly with egg wash. Top with a second round and press the edges firmly with a fork to seal. Poke a small steam vent in the top of each cookie with the tip of a knife.
Brush the tops with remaining egg wash and sprinkle with the cinnamon-sugar mixture. Bake at 375°F (190°C) for 16–18 minutes until the edges are deep golden and the tops look crisp. Cool on the baking sheet for 5 minutes, then transfer to a wire rack. The cookies firm up as they cool and the filling settles.