In a medium sauce pan, heat the sugar, cornstarch, milk and cream over medium heat. Whisk this mixture constantly as it heats, until it is thickened and bubbling. Pass through a fine mesh strainer in to a bowl and cover with plastic wrap directly on the top of the mixture. Allow to cool and place in the fridge overnight, covered in plastic wrap. Make sure your ice cream bowl is in the freezer at least 24 hours (longer is better) before you will churn the ice cream.
Place the strawberries in blender and puree until very smooth. Strain through a fine mesh strainer into a mason jar or into a bowl covered with plastic wrap and put this into the fridge overnight as well. You want everything cold before you churn.
Pour both mixtures into your ice cream bowl the next day and churn according to the instructions for whatever ice cream maker you use. When it is soft serve consistency, scrape into your container and freeze for at least a few hours before eating. I also freeze my ice cream container ahead of time so the mixture isn't going into a warm container. See my post for the ice cream containers I use and love.