In a medium saucepan, melt the butter over medium-high heat. Add the apples and cook until they start to soften, 3 to 4 minutes. Stir in the brown sugar and lemon juice, bring to a simmer and cook until the apples are soft and the liquid is starting to reduce, 3 to 4 more minutes. In a small bowl, whisk together the apple juice and cornstarch and add it to the pan. Cook, stirring, until the mixture thickens, about 1 more minute. Remove from the heat and let cool.

Preheat the oven to 350˚F. Line a baking sheet with parchment paper. Roll out 1 piece of pie dough into a 12-inch round. Using a 3-inch cutter, cut out 12 small rounds and arrange on the baking sheet. Re-roll the scraps and cut out 2 more rounds. Roll the other piece of dough into a 12-inch round. Cut into ½-inch-wide strips. Cut the strips into shorter 3 ½- to 4-inch pieces (you should have about 56 pieces).

Arrange the dough rounds on the baking sheet. Place 1 tablespoon of the apple mixture in the center of each. Lay 2 parallel strips of dough on top, then lay 2 more strips perpendicular to the first to look like a lattice. (Alternatively, weave the strips into a lattice.) Trim the edges of the strips, then press around the edges with a fork to seal. Bake until golden, 18 to 20 minutes. Remove to a rack to cool.

For the glaze: In a small bowl, whisk together the powdered sugar, heavy cream, and cardamom. Drizzle over the cookies.