Prepare the apple filling: In a medium saucepan, combine the diced apples, granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, and water. Cook over medium heat, stirring occasionally, until the apples are softened and the filling has thickened, about 10-15 minutes. Remove from heat and let cool slightly.
Make the biscuit dough: In a large bowl, whisk together the flour, baking powder, and salt. Cut in the cold butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs. Stir in the cold milk until just combined. Do not overmix.
Assemble the biscuits: Turn the dough out onto a lightly floured surface. Gently pat the dough into a rectangle about ½ inch thick. Cut the dough into squares or circles using a biscuit cutter or a knife.
Fill the biscuits: Place a spoonful of the apple filling in the center of each biscuit. Gently press the edges of the dough around the filling to seal.
Bake: Place the biscuits on a baking sheet lined with parchment paper. Bake at 400°F (200°C) for 25-30 minutes, or until golden brown.
Make the glaze (optional): While the biscuits are baking, whisk together the powdered sugar, milk, and vanilla extract until smooth. Drizzle the glaze over the warm biscuits after they have cooled slightly.
Serve: Serve the apple pie biscuits warm.