Combine the sugar and water in a small saucepan and stir over medium heat until the sugar dissolves. Remove the simple syrup from the heat and let cool.
Combine the lemon juice and 1 cup simple syrup in an 8-inch square baking dish. Move to the freezer and chill, raking the mixture with a fork every 30 minutes, until flaky and frozen, about 3 hours total.
Pour the frozen mixture into a blender and puree with the water. Divide among six 4-ounce ramekins or disposable paper cups and freeze until firm, at least 4 hours or up to 2 days. Remove from the freezer 15 to 20 minutes before scooping and slurping.