Place the butter in a large skillet. Melt over medium heat while you assemble the apple mixture.
In a large bowl, stir together the apples, brown sugar, flour, and cinnamon until combined and apples are coated.
Add the apple mixture to the melted butter. Cook over medium heat, stirring frequently, until the mixture thickens (about 10 minutes).
Transfer the apple mixture to a 3-quart baking dish.
To make the topping:
Preheat oven to 400°F (200°C).
Combine the flour, brown sugar, baking powder, baking soda, and salt. Add the butter and mix using a pastry blender or a fork until the largest bits of butter are about the size of small peas.
Add the milk and sour cream, stirring just until combined. Use your hands if necessary to get the dough mixed with minimal working of the dough. Dough will be sticky.
Drop the dough in 2-tablespoon portions over apple mixture. Sprinkle the tops of the dough evenly with cinnamon sugar.
Bake 25 to 30 minutes, or until the topping is lightly browned and the filling is bubbly.