Blend doughnut sticks in a food processor until finely ground. Combine ground doughnuts with melted butter in a bowl. Press mixture into the bottom of an 8x8-inch glass baking dish. Freeze crust for 10 minutes.
Spread ice cream over the frozen crust. Place back into the freezer for about 5 minutes.
Meanwhile, combine milk, pudding mix, and instant coffee until smooth. Fold in whipped topping and spread over the coffee ice cream in the crust. Cover with plastic wrap and freeze for 2 hours. Place mini doughnuts on top before serving.