Preheat oven to 350 degrees F (175 degrees C). Grease and flour one 8 x 8 inch square pan.
Sift the flour, sugar, 3 tablespoons cocoa, baking powder, baking soda and salt together. Make a well in the center and add the water, oil, vinegar and vanilla. Beat until smooth and pour batter into the prepared pan.
Bake at 350 degrees F (175 degrees C) for 30 minutes.
To Make Frosting: Combine the 1/2 cup cocoa powder, sweetened condensed milk and butter in the top half of a double boiler and cook until thick. Spread warm mixture over the top of a slightly cooled cake.