Grease an 8x8x2-inch baking pan. Set the baking pan aside.
In a medium saucepan, melt butter and unsweetened cocoa powder over low heat, stirring occasionally. Remove the mixture from the heat.
Stir in the eggs, sugar, and vanilla. Using a wooden spoon, beat lightly until just combined. Stir in flour and nuts. Spread butter in prepared baking pan.
Bake in a 350 degree F oven 25 minutes. Cool completely in pan on a wire rack and spread with frosting.
In a mixing bowl, mix melted butter,unsweetened cocoa powder, milk, and vanilla. Add sifted powdered sugar and beat with an electric mixer until the frosting is spreadable.