In a small saucepan bring the half-and-half and heavy whipping cream to a scalding point on medium low heat. “Scalding” means gently heating without allowing to reach boiling point. Immediately remove from heat.
In a bowl (or stand mixer), whisk cream cheese, egg yolks and sugar till they turn light in color and fluffy.
Once the egg yolks mixture is done, gradually pour and mix the scalding half-and-half & heavy whipping cream mixture in batches. Keep whisking constantly to avoid the eggs from curdling.
Now place the bowl over a saucepan of simmering water, keep stirring continuously with a spoon until it starts to thicken up. Once the consistency is thick enough to coat the back of the spoon, the custard is ready. Immediately remove the bowl from the heat and stir it for few minutes to help it to cool down a bit.
Meanwhile, make strawberry puree, chop up strawberries, and crush the honey graham crackers.
Now add the vanilla essence and strawberry puree and stir well one last time.
Cover the custard and let it chill in the refrigerator for few hours before making the ice cream.
Once you're ready to make the ice cream, transfer the chilled custard to the chilled container of your ice cream machine and process according to the manufacture's instruction.
During the last 5 minutes of ice cream churning in the machine, add the chopped strawberries and graham crackers.
Transfer the ice cream to a container and let it sit in the freezer to set, about 3-4 hours. Serve Chilled.