Step 1Place the diced persimmons in a food processor or blender and pulse or blend to puree. Pass the puree through a sieve into a bowl, pressing on it with the back of a spoon.
Step 2Combine the simple syrup and lemon juice. Add to the puree and mix well.
Step 3Freeze the puree in 2 batches in an ice cream maker according to the manufacturer's directions. This will be a very soft, creamy sorbet.
You can make a granita if you don't have an ice cream maker. After pureeing the fruit and adding the syrup and juice, pour the mixture into a shallow metal pan. Freeze, scraping the mixture with a fork every 30 or 40 minutes, until somewhat firm, 3 to 4 hours.