Assemble drip coffee maker according to manufacturer's directions. Place ground coffee in the coffee filter or filter basket. Add 1 3/4 cups water to coffee maker and brew; set coffee aside.
Combine sugar and 1/3 cup water in a small saucepan. Bring to a boil, and cook 1 minute or until sugar dissolves. Stir in vanilla and cinnamon. Remove from heat, and stir in brewed coffee and milk.
Cool coffee mixture completely; pour into an 8-inch square baking dish. Cover and freeze at least 8 hours or until firm. Remove coffee mixture from freezer; scrape entire mixture with a fork until fluffy. Spoon into a freezer-safe container; cover and freeze for up to 1 month. Garnish with mint sprigs, if desired.