Meanwhile, put strawberries, basil, and lemon juice in a food processor and pulse 5 or 6 times-- until finely chopped. Combine strawberry mixture into the chilled milk mixture.
Pour milk mixture into freezer container of a 1-qt. electric ice-cream maker, and freeze according to the ice cream maker's instructions (this may vary).
Remove ice cream container from ice-cream maker, and freeze 30 minutes.
Transfer mixture to an airtight container or pan covered tightly with tin foil; freeze 3 to 4 hours or until firm.