Preheat oven to 350°F and grease and flour two 8-inch cake pans.
Sift cocoa, flour, baking soda, salt and sugar into a medium bowl.
Combine mayonnaise, vanilla and water in an electric mixer at low speed. Once smooth, slowly add in dry mixture until just incorporated (do not over beat).
Pour batter into cake pans and bake for 30 minutes or until done.
Cool on wire racks.
For icing, In the bowl of an electric mixer, beat butter and cocoa with whisk attachment until smooth. Add vanilla. Add confectioners' sugar alternately with milk until icing reaches a spreadable consistency.
Spread a portion of the icing on top of one layer of cake. Stack the other layer on top. Use remaining icing to cover the top and sides.
Notes
Note: You may have a hard time frosting the sides of the cake without crumbs showing. If so, spread a thin layer of frosting on sides, refrigerate for 15 to 20 minutes and then do a second thicker coat.