I never had Applesauce Cake, but I found this recipe online. Personally, I would leave out the cloves and allspice (not a fan) but the picture looked like what you described. Enjoy!
2 1/2 cups all-purpose flour
2 cups sugar
1 1/2 teaspoons baking soda
1/4 teaspoon baking powder
1 1/2 teaspoons salt
3/4 teaspoon cinnamon
1/2 teaspoon clove
1/2 teaspoon allspice
1/2 cup vegetable oil
1/2 cup water
1 1/2 cups unsweetened applesauce
2 eggs
1 Preheat oven to 350 degrees.
2 Grease a 9×13 baking dish.
3 Combine all ingredients; pour in prepared pan.
4 Bake 60-65 minutes or until done.