Melt the butter in a medium skillet over medium heat.
Add the diced apples, brown sugar, cinnamon, nutmeg, salt, and lemon juice. Cook for 5 to 7 minutes, stirring occasionally, until the apples begin to soften.
Stir together the cornstarch and water in a small bowl. Pour the mixture into the skillet and continue cooking for 1 to 2 minutes, until the liquid thickens and coats the apples.
Remove the skillet from the heat and stir in the vanilla. Transfer the apple filling to a bowl and allow it to cool completely. The filling should be thick and no longer warm before assembling the cookies.
Preheat the oven to 375°F. Line a large baking sheet with parchment paper.
Unroll the first pie crust onto a lightly floured surface. Use a 2¾- to 3-inch round cookie cutter to cut approximately 10 circles, gently rerolling the scraps once as needed. Arrange the circles on the prepared baking sheet.
Unroll the second pie crust and cut it into ¼- to ½-inch-wide strips. Weave the strips together on a lightly floured sheet of parchment paper to create a lattice crust. Cut approximately 10 lattice circles using the same cookie cutter.
Spoon about 1 level tbsp of cooled apple filling into the center of each bottom crust, leaving approximately ¼ inch of uncovered crust around the edge. Avoid overfilling the cookies.
Place one lattice circle over each portion of filling. Press the edges together, then seal them firmly with the tines of a fork.
Whisk the egg with 1 tsp water. Lightly brush the tops of the cookies with the egg wash and sprinkle with coarse sugar.
Bake for 16 to 20 minutes, until the crust is crisp and golden brown. Begin checking the cookies at 16 minutes since baking time can vary by pie-crust brand and cookie size.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack.
Once the cookies have cooled for at least 15 minutes, drizzle them with salted caramel sauce. Serve slightly warm or at room temperature.