Preheat oven to 350°F and grease a 9x9 or 8x8 square baking pan, or line with parchment paper.
In a large bowl, beat together sugar, applesauce, butter, egg and vanilla.
In another bowl, mix together flour, baking soda, cinnamon, ginger, baking powder, salt, and allspice. Beat into wet ingredients until well blended. Mixture won't be completely smooth due to the applesauce.
Pour batter into pan and spread evenly to the edges. Bake for 25-30 minutes, until golden brown and a tester inserted into the center comes out clean or with only moist crumbs. Let cool.
To make the frosting, first melt the butter in a small saucepan over medium heat, stirring with a silicone spatula. Continue to cook, stirring, as the butter starts to bubble and foam. The bubbles will get large, then start to dissipate, as the butter browns. As soon as you notice the color change, stir well and remove from heat. Transfer to a mixing bowl and let cool completely.
Once the butter is room temperature, add the powdered sugar and beat with a hand mixer. It will form small pellets. Add the vanilla and milk and continue beating until frosting is smooth, adding enough milk for the texture you prefer.
Frost the cooled cake and serve. Store leftovers in an airtight container.