Preheat oven to 350 degrees. Coat a 10-inch angel-food-cake pan with cooking spray; set aside. Make crumb topping: Stir together 4 tablespoons butter, the dark-brown sugar, oats, 3/4 teaspoon cinnamon, and 1/4 teaspoon salt until smooth. Stir in 1/2 cup pecans; set aside.
Make batter: Sift flour, baking soda, nutmeg, cloves, and remaining 2 teaspoons cinnamon and 3/4 teaspoon salt; set aside. Put remaining 8 tablespoons (1 stick) butter, the granulated sugar, and light-brown sugar in the bowl of an electric mixer fitted with the paddle attachment. Cream on medium speed until smooth, about 3 minutes. Mix in the eggs, one at a time. Reduce speed to low; mix in applesauce and then flour mixture. Stir in remaining cup nuts by hand.
Pour batter into oiled pan; sprinkle reserved crumb topping over batter. Lay apples on top, tucking some into batter.
Bake until a cake tester inserted near center comes out clean, about 1 hour and 10 minutes. Let cool in pan on a wire rack.
Cake can be stored at room temperature, covered with plastic wrap, up to 3 days.