1. Combine chocolate milk, bourbon, sugar, vanilla, and nutmeg in a 2-qt. measuring cup, stirring until sugar dissolves. Pour half of mixture into paper cups; cover and chill remaining half.
2. Cover each cup with aluminum foil; make a small slit in center, and insert 1 ice-cream spoon into each cup. Freeze 4 hours or until firm.
3. Carefully remove foil from cups. Pour 1 Tbsp. chocolate syrup into each cup; freeze, uncovered, 2 hours. Pour remaining milk mixture into cups; freeze, uncovered, 8 hours or until firm.