Preheat the oven to 350F. Lightly grease an 8-inch square pan with cooking spray.
In a large bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, nutmeg, and allspice.
In a separate bowl, whisk together the sugar, brown sugar, apple sauce, melted butter, milk, egg and vanilla extract until well-blended.
Combine the wet and dry ingredients, stirring until no streaks of flour remain. Pour into prepared pan, and bake on the center rack for 33-36 minutes, until the edges of the cake just start to pull away from the sides of the pan. They’ll pull away further as it cools. Let it cool while you make the frosting.
Meanwhile, make the frosting: melt the butter in a small saucepan over medium-low heat. Stir occasionally until the foaming stops, and let it cook until it turns a golden brown color and smells nutty. Stir occasionally to prevent any burning.
Pour the melted browned butter over the powdered sugar, and whisk to combine. Add the vanilla and salt. (If the frosting doesn’t come together, add a splash of milk or cream to thin it out).
Frost the cooled cake, and slice it into 16 equal pieces.