Heat oven to 350° F. Cream butter or margarine, sugar, and egg. In another bowl dissolve soda in applesauce; add to creamed mixture. Beat in remaining ingredients and blend until smooth. Batter should be fairly thick. Pour into a greased 8-inch square baking pan and bake for 35 to 40 minutes, or bake in a greased 8 1/2 x 4 1/2 x 2 1/2 -inch loaf pan for 50 to 60 minutes. Cool; prepare the following frosting.
Browned Butter Frosting:
*In a small saucepan, melt butter over low heat. Cook butter slowly until it is browned, but not burned. Stir in cream, vanilla, and confectioners' sugar. Blend until smooth. Frost cooled cake.