Make the strawberry syrup:
In a blender, combine the strawberries, sugar, and water. Puree until smooth then transfer to a small saucepan. Bring to a boil over medium/high heat and then reduce the heat to medium. Allow to cool until the mixture has a syrupy texture, about 10 minutes. Remove from heat and skim the foam that has accumulated on the surface of the syrup. Set aside to cool.
Make the strawberry quick jam:
In a large saucepan, combine strawberries, sugar, and vanilla bean paste. Bring the mixture to a boil over medium/high heat stirring constantly. Reduce the heat to medium and continue cooking, stirring, until the strawberries have softened and the liquid has reduce, about 15 minutes. Remove from heat and allow to cool completely. While the jam is cooling, make the cornstarch slurry and ice cream base.
Make the ice cream base:
In a small bowl, combine the cornstarch and milk. Mix until the cornstarch is completely dissolved and set aside.
Prepare a large bowl full of ice water. In another large bowl (one that can fit inside the bowl of ice water) add the cream cheese and salt. Mix until combined and set aside.
In a medium saucepan combine he milk, heavy cream, vanilla extract, corn syrup, and sugar. Stir until combined and bring to a boil over medium heat. Be sure to watch it so it doesn’t spill over. Reduce the heat to medium/low and add the cornstarch slurry. Whisk the mixture vigorously until slightly thickened, about 3-5 minutes. Remove the mixture from the heat and whisk into the large bowl with the cream cheese mixture. Whisk until the cream cheese is completely combined. Place the ice cream base in the bowl of ice water and allow to cool completely.
Once completely cooled, mix in the strawberry syrup, strawberry jam, and rum. Transfer the mixture to a 1 gallon Ziploc bag and allow it to steep in the refrigerator for at least 4 hours or preferably
overnight.
Freeze in an ice cream maker according to manufacturer’s instructions and transfer ice cream to a container with a tight fitting lid. Place in the freezer. Try not to eat it every 5 minutes. Oh, who the hell am I kidding?! Pop in a movie and spoon that sucker to your heart’s content.
Makes about 1.5 quarts of ice cream.