In a medium saucepan over medium-high heat, bring milk and half-and-half just to a boil. Remove from heat. In a stand mixer fitted with the paddle attachment, combine sugar, egg yolks, and cream cheese on high-speed until smooth. Remove bowl from mixer and gradually whisk in the hot milk mixture ? go very slowly so you don?t cook the egg, whisking constantly. Transfer mixture back to the saucepan and cook over medium-low until it reaches 160°, about 5 minutes. Stir constantly. Remove from heat and place pan in an ice-filled bowl. Cool completely, stirring occasionally. As the milk mixture cools, combine strawberries, confectioners? sugar and water in a small saucepan. Bring to a boil. Reduce heat and simmer for 10 minutes, stirring frequently. The strawberry mixture should thickened slightly. Remove from heat and cool completely. When the strawberry and milk mixtures have cooled, stir the strawberry mixture into the milk mixture until thoroughly combined. Refrigerate 2-4 hours until very cold.
Make the ice cream: When mixture is very cold, pour into the freezer can of ice cream maker and freeze according to manufacturer?s instructions.
For those with the ice cream bowl for a stand mixer, set the mixer on speed 1 (stir) and gradually pour mixture into mixer. Continue mixing for 15-20 minutes, or until mixture has reached desired consistency.
Freeze: Pour into a plastic container and place in freezer. Ice cream will take at least 4 hours to freeze completely.