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Add the lemons, sugar, honey and 2 1/2 cups water to a food processor and puree until smooth, about 2 minutes. You may have to do this in batches to fit into your food processor. (If you own a powerful blender such as a Vitamix you may omit this step).
Pour the puree into a blender and continue pureeing on the highest setting for another 2 minutes. Strain the mixture through a fine mesh sieve. Use a spoon to squeeze any excess juice out of the pulp by pressing it against the side of the strainer. Pour the puree into the canister of an ice cream machine and process according to the manufacturer's instructions.
For a softer texture serve immediately, or transfer the frozen lemonade to a container and freeze until hard.