Preheat oven to 350F. Grease a 13-by-9-inch baking pan.
To make cake, combine sugars, flour, cocoa, baking powder, baking soda and salt. Add eggs, milk, oil and vanilla; whisk vigorously until smooth. Stir in boiling water. Pour into prepared pan. Bake 35 minutes or until toothpick inserted in center comes out clean. Cool.
To make frosting, mix butter, powdered sugar and vanilla with a mixer; beat well. Add peanut butter. Add 2 tablespoons milk slowly, beating until very creamy. Add up to 3 additional tablespoons milk if necessary to achieve desired consistency. When cake has cooled, frost top.