In a saucepan, combine the apples, 1 stick of the butter, 1 cup of the sugar, the vanilla extract, 1 teaspoon of the salt, the cinnamon and nutmeg. Cook until the apples become soft, 15 to 20 minutes.
In a separate bowl, combine the flour, baking powder, remaining 1 cup sugar and remaining 1 teaspoon salt. Then add the milk and mix until all of the dry ingredients are moistened.
Meanwhile, preheat the oven to 350 degrees F.
Add 1 tablespoon of butter to each skillet or ramekin. Place the skillets on a cookie sheet and heat them in the oven until the butter begins to bubble, about 10 minutes. Carefully remove the cookie sheet with the skillets, and spoon the batter until just below the rims (about half full if using a ramekin). Immediately spoon the apple mixture over the top of the batter. Place in the oven and bake until golden brown, about 20 minutes.