To prepare filling: Melt butter in a 10-inch cast-iron skillet over medium heat, add apple slices and cook, stirring frequently, for about 15 minutes. Mix flour, sugars, cinnamon, nutmeg and salt in a bowl and gently stir mixture into apples. Continue to cook until apples are soft but not mushy, about 5 minutes.
Mix cornstarch and 1 1/2 cups water together in a small bowl, and stir into apple mixture. Set aside.
To prepare topping: Mix together flour, 1/4 cup sugar, baking powder, and salt in a large bowl. Using 2 knives or a pastry blender, cut in the butter until the mixture resembles coarse crumbs. Add the milk and vanilla, and quickly blend together with spatula.
To assemble the pie: Drop heaping tablespoons of biscuit mixture on top of apples, covering most of the center of the mixture. Sprinkle top with remaining sugar and bake in preheated 350-degree oven for 25 to 40 minutes, or until top is golden. Remove pie from oven and let stand for at least 5 minutes before serving.