Preheat the oven to 350 degrees. Spray a 9 x 13 baking dish with baking spray and set aside.

In a large bowl beat the butter on medium speed with a mixer for 1-2 minutes. Add in the brown sugar and granulated sugar and beat for 2 minutes.

Add in the eggs and mix on medium speed for 1 minute. Mix in the vanilla.

In a medium bowl combine the flour, cinnamon, baking powder, salt, allspice, and nutmeg. Mix well.

Add one third of the flour mixture to the butter mixture and mix on low speed. Add in half the milk and mix on low speed until just combined. Add another third of the flour mixture and mix on low until combined. Add the remaining milk and mix on low until just combined. Finish up by adding in the remaining flour and raising the speed to medium and mix until everything is combined.

Fold in the diced apples.

Pour the batter into the prepared pan. Bake for 40-45 minutes or until a toothpick inserted into the center comes out clean.

Remove from oven and cool completely on a wire rack.

To make the frosting set a heatproof bowl over a saucepan with a few inches of water that is simmering over medium heat. Add the brown sugar, egg whites, and cinnamon and stir with a rubber spatula. Continue stirring frequently until the sugar melts, about 8 minutes.

Remove the bowl from the pan and beat with a mixer on medium speed for 2-3 minutes. Increase the speed to medium high and beat until stiff glossy peaks form, about 3 minutes.

Add the butter 2 tablespoons at a time, beating on medium high speed until incorporated, before adding more butter.

Spread the frosting over top of the cake. Decorate with cinnamon or fall sprinkles.