In a large bowl, mix strawberries, honey, sugar, and lemon juice. Allow to sit 15-20 minutes, or until the strawberries have released their juices.
Mash the mixture with a potato masher, fork, or pastry cutter until there are no longer any large strawberry pieces (if you prefer a smooth ice cream, use a blender or food processor instead*).
To the strawberries, mix heavy cream, half and half, and vanilla extract. Stir just until combined.
Pour the mixture into your ice cream maker, and follow your manufacturer's instructions (mine took about 15-20 minutes to churn). You can eat the ice cream soft-serve, or freeze it in an airtight container for 3-4 hours to thicken it up slightly. For best results, allow the ice cream to sit 5-10 minutes to soften before serving.