Whisk together your cream cheese and your sugar until smooth.
Add in the heavy cream and milk and whisk until mostly combined. **
Pour mixture into your prepared ice cream mixer. Allow to churn according to manufactures instructions.
Scoop 1/3 of the ice cream mixture into a 3QT pan (a bread pan is perfect for this)
Scoop 1/3 of the strawberry puree, and sprinkle 1/3 of the graham cracker crumbs over the top of the ice cream.
Swirl together with a knife. (Swirl well, any thick strawberry sauce areas will become a bit icy)
Repeat with the second third, and final third of the ice cream, sauce and graham crackers.
Cover with plastic wrap and place in your freezer to harden for 6 hours, up to overnight.