Heat oven to 350 degrees.
Butter two 8-inch cake pans, and line base of each with parchment paper.
In food processor, combine 1/2 cup walnut pieces and sugar.
Process to a fine powder.
Add butter, flour, baking powder and eggs.
Process to a smooth batter.
Add 2 tablespoons milk to coffee mixture and pour this down feed tube with motor running.
Mixture should be just soft enough to drop from a spoon; if not, add more milk.
Divide cake batter between two pans.
Bake until risen and springy to touch, about 25 minutes.
Cool on a rack for 10 minutes, then remove from pan and place on rack.
Remove parchment paper.
Cool completely.
For frosting: Place confectioners' sugar in food processor and pulse until lump-free.
Add butter and process until smooth.
Add coffee mixture down feed tube and pulse until well blended.
Place one cake upside down on a plate or cake stand.
Spread with about half the frosting.
Place second cake right side up on frosting, and cover top and sides in a swirly pattern.
Place a walnut half in center of cake, and gently press remaining halves into top of cake, around the edge.