Preheat oven to 350° F.
In a medium sized bowl, stir together flour, baking powder, salt, granulated sugar, and spices. Set aside.
In a smaller bowl, stir applesauce, milk, melted butter, vanilla, and grated carrot together to combine. Pour wet ingredients into dry ingredients and mix to create a thick batter.
Pour into a small 8-inch rectangular casserole dish with high sides.
In a separate bowl, stir sugar, brown sugar, and walnuts together. Spread over the top of the batter evenly.
Pour hot water over the entire thing (without stirring!) and bake for 40-50 minutes or once the middle is set. Be sure to place on a baking sheet in case it bubbles over.
Cool 5-10 minutes before serving. Serve with more nuts and cheesecake ice cream.