Combine the first 3 ingredients in an 8-inch baking dish, stirring until sugar and espresso dissolve. Cover and freeze for 45 minutes. Stir espresso mixture with a fork every 45 minutes until completely frozen (about 4 hours).
Remove espresso mixture from freezer; scrape mixture with a fork until fluffy. Combine ice cream, whipped topping, and liqueur. Serve over espresso ice.