Preheat the oven to 350 degrees. Prepare a large-rimmed baking sheet with parchment paper.
In a medium saucepan over medium-high heat, melt the butter.
Add the apples and cook until they begin to soften - about 3 to 4 minutes.
Add the brown sugar and lemon juice. Bring the mixture to a boil, reduce the heat to medium-low, and continue cooking until the apples are soft and the liquid starts to reduce approximately 3 to 4 more minutes.
In a small bowl, whisk together the water and cornstarch. Add this mixture to the saucepan and cook while stirring, until the mixture thickens, about 1 minute. Remove the saucepan from the heat and allow the mixture to cool.
Roll out one piece of pie dough into a 12-inch round. Using a 3-inch cutter, cut out 12 to 14 rounds and arrange them on the baking sheet. Re-roll the dough scraps and cut out additional rounds.
Place about 1 tablespoon of the apple mixture in the center of each round.
Roll the other piece of pie crust dough into a 12-inch round. Cut it into 1/2-inch wide strips. Cut the strips into shorter pieces to match the width of the cookies.
Place 4 strips of dough on top of each cookie with 2 strips running parallel over the cookie and the other 2 perpendicular to those strips. Alternate an over-under pattern to create a lattice.
Trim the edges of the strips where they overhang the cookie edges, then press around the edges with a fork to seal.
Bake for about 18 to 20 minutes or until the cookies turn golden. Transfer the cookies to a wire cooling rack.