Combine the sugar with 3 cups of water in a small saucepan. Bring to a simmer, stirring to dissolve the sugar. Remove from the heat and add the lemon juice. Pour the mixture into 3 ice cube trays and set aside to cool to room temperature, about 20 minutes. Place the trays in the freezer to harden, 4 to 8 hours depending on your freezer.
When ready to serve, add the ice cubes and the lemon peel to a food processor and pulse until the ice is fluffy and broken down. Serve with a straw and a spoon, and be prepared to be whisked away to the summer streets of Chicago.