Squeeze the juice from the lemon half into a medium bowl through a strainer or your fingers to remove the seeds. Add 1/4 cup sugar. Stir to dissolve.
Fit the food processor with the grating disk and turn it on. Push the frozen fruit through the feed tube a few pieces at a time. When it’s all grated, transfer to the bowl with the sweetened lemon juice.
Stir gently with a fork to combine. Taste and add more sugar if necessary. Serve right away.