Preheat oven to 350 degrees F (175 degrees C). Grease and flour two 9-inch round pans, and line them with wax paper or parchment.
Sift flour with baking soda. Dissolve 1/3 cup instant coffee in the water.
In a large bowl, beat white sugar, 3 eggs, and vanilla until light and fluffy. Blend in mayonnaise and the melted chocolate. Add flour mixture alternately with the coffee to the batter, one-third at a time; mix with electric mixer on low speed after each addition. Pour batter into prepared pans.
Bake in preheated oven until cake tester inserted in center comes out clean, about 35 minutes. Cool cake in pans for 10 minutes. Remove, and cool completely on wire racks.
Melt 4 squares chocolate with milk and 2 tablespoons instant coffee in a double boiler; blend until smooth. Pour hot chocolate mixture into the beaten egg, whisking constantly. Allow to cool.
In a mixing bowl, cream butter and confectioners' sugar until light and fluffy. Add chocolate mixture to butter mixture, and blend frosting well.
When layers have cooled, cut each horizontally to make 4 layers. Fill and frost the cake with the frosting.