Preheat oven to 350 degrees F.
Cake: In a mixer with a whip attachment, mix the dry ingredients together. Add the softened butter, lukewarm milk, beaten egg, and vanilla and mix well for 2 to 3 minutes. Pour into a parchment lined 8 by 8-inch cake pan. Bake at 350 degrees F for 25 to 35 minutes.
Broiled Icing: In a mixer with a paddle attachment, mix all the ingredients together. When the cake is done, take it out of the oven and, while it's warm, spread the icing on the cake to cover the top. Place it under the broiler until it bubbles all over and starts to brown. Cool then cut into squares and serve.