In a large bowl with a hand mixer, combine sugar, cream cheese, and egg yolks and beat at high speed until smooth.
In a medium heavy saucepan, combine milk and half-and-half. Bring the mixture just to a boil and then remove the pan from the heat.
Add half of the hot milk mixture to the sugar/cream cheese/egg mixture, while stirring constantly with a whisk to combine. Return this milk/sugar/cream cheese/egg mixture back to the saucepan with the remaining milk in it.
Cook over medium-low heat for about 5 minutes, stirring the whole time, until the mixture reaches 160 degree when tested with a thermometer or until it coats the back of a wooden spoon.
Pour mixture into a heat-proof bowl and place the bowl in an ice bath (another larger bowl filled with ice) or in the fridge so it can cool.
While ice cream is cooling, prepare the blueberry sauce. In a small saucepan, combine blueberries, confectioners' sugar, and water. Bring to a boil and then reduce the heat and simmer for about 10 minutes, stirring frequently. Remove saucepan from heat and let mixture cool.
When milk and blueberry mixtures are both cooled, stir blueberry mixture into milk mixture. Place mixture in fridge to chill for at least 3 hours.
Process ice cream in ice cream maker according to manufacturer's instructions. When the ice cream is almost done processing, add the graham cracker pieces in.
Scoop ice cream into airtight containers and let chill in freezer for another 2 hours.