In a medium saucepan over medium heat, melt the butter. Add the finely chopped apples, brown sugar, granulated sugar, cinnamon, nutmeg, and a pinch of salt. Stir well to coat everything.

Cook the apple mixture, stirring occasionally, for 5–7 minutes, until the apples start to soften and release their juices. They should be tender but not mushy.

In a small bowl, whisk the cornstarch and water until smooth. Pour the slurry into the apples while stirring constantly. Cook for 1–2 more minutes, until the filling is glossy and thick, like a chunky jam.

Remove the pan from the heat and stir in the vanilla extract. Spread the filling onto a plate or shallow dish to help it cool completely before assembling the cookies.

Preheat the oven to 375°F (190°C). Line 2 baking sheets with parchment paper or silicone baking mats.

Lightly flour your work surface. Unroll one pie crust and, if needed, gently roll it to an even 1/8-inch thickness. Repeat with the second crust.

Using a round cookie or biscuit cutter (about 2 1/2 to 3 inches in diameter), cut as many circles as possible from the dough. Gather the scraps, gently knead, reroll, and cut more circles. You should get about 36–40 circles total for 18–20 cookies.

Place half of the dough circles on the prepared baking sheets. In a small bowl, beat the egg with 1 tablespoon of water. Lightly brush just the edges of each dough circle with the egg wash.

Spoon about 1 to 1 1/2 teaspoons of cooled apple filling into the center of each dough circle, leaving a clean border around the edge. Do not overfill to prevent leaking.

Place a second dough circle over each filled bottom. Gently press the edges together with your fingers, then crimp all around with a fork to seal tightly and create a hand-pie look.

Use a sharp knife to cut a small X or a couple of tiny slits in the top of each cookie to vent steam. Brush the tops with more egg wash and sprinkle with coarse sugar.

Bake one sheet at a time for 14–18 minutes, or until the cookies are golden brown and crisp on the edges. Rotate the pan halfway through if needed for even browning.

Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely. They will crisp slightly as they cool.

In a small bowl, whisk together the powdered sugar, 2 tablespoons of milk, vanilla, and cinnamon until smooth. If needed, add more milk a few drops at a time until the glaze is thin enough to drizzle.

Once the cookies are completely cool, drizzle the cinnamon glaze over the tops with a spoon or piping bag. Let the glaze set for 20–30 minutes before stacking or storing.