In a medium-sized mixing bowl, beat together the butter, apple sauce and sugars until smooth. Add the eggs, beating to combine. Stir in the baking powder, baking soda, nutmeg, salt and vanilla.
1/4 cup butter, 1/4 cup apple sauce, 1/2 cup sugar, 1/3 cup brown sugar, 2 each eggs, 1 1/2 tsp baking powder, 1/4 tsp baking soda, 1/2 tsp ground nutmeg, 3/4 tsp salt, 1 tsp vanilla extract, 2 2/3 cups all-purpose flour, 1 cup milk