In a small saucepan, heat the apples, 1/3 cup white sugar, 1/4 tsp cinnamon, nutmeg, cornstarch and water on low heat for 5-7 minutes, or until the apples a bit soft. Remove from heat.
In a small bowl, mix together the brown sugar and 1/2 tsp cinnamon. Set aside.
Preheat the oven to 350°F. Lightly grease a 9x13 baking pan. Set aside.
In a medium bowl, cream together the butter and 3/4 cup white sugar. until light and fluffy. Add in the applesauce and vanilla. Mix in the eggs one at a time, beating after each addition.
In a separate bowl, mix together the flour, baking powder, baking soda, salt and cinnamon. Slowly add in the dry ingredients and the sour cream, alternating between the two, until fully combined.
Pour half of the batter into the prepared baking pan. Spread the apples over the batter, and sprinkle 2/3 of the brown sugar mixture on top. Pour the remaining batter on top and sprinkle the remaining brown sugar on top.
Bake in the preheated oven for 45-50 minutes, or until a toothpick inserted in the centre comes out clean.