Bring sugar and 3/4 cup water to a boil in a saucepan, stirring constantly; remove from heat. Add pepper halves; let stand 1 hour. Discard pepper halves.
Process cucumber in a blender until smooth. Add sugar syrup, milk, and next 5 ingredients; process until smooth. Pour into freezer container of a 4-quart electric freezer.
Freeze according to manufacturer's instructions. Serve immediately, or store in freezer. (If stored in freezer, let stand at room temperature 10 to 15 minutes before serving.) Garnish, if desired.