In a small saucepan, heat cream, milk, cornstarch and agave on medium-low heat until bubbly and thickened slightly, whisking constantly, about 5 minutes. Once the mixture is thick enough to coat the back of a wooden spoon, place the pot with hot mixture into a large bowl filled with ice water and whisk it constantly until cool to the touch.

Place strawberries in a blender and puree for about two minutes until completely smooth. Push mixture through a fine mesh strainer over a bowl to remove seeds and discard them. If you don't mind the seeds, you can omit this step.

Once the cream mixture is cooled down, whisk in strawberry puree until fully incorporated. Cover mixture with plastic wrap and place in refrigerator to chill for two hours. Pour mixture into ice cream maker and freeze according to manufacturer's instructions, about 20-30 minutes or so. Pour into an airtight container and freeze 4-6 hours or overnight. Allow ice cream to set out for a few minutes before scooping.