Prepare an ice bath. Fill a large bowl with ice and place a smaller bowl on top of the ice. Add the heavy cream and vanilla to the bowl and set aside.
In a small bowl, mix 2 tablespoons of milk with the cornstarch until smooth.
Pour the remaining 2 cups of milk, sugar and corn syrup in a medium saucepan. Bring to a rolling boil over medium heat. Don’t leave the stove! Be sure to watch that the milk doesn’t boil over. Simmer the mixture for 3-4 minutes. Remove from the heat and whisk in the cornstarch mixture.
Return the saucepan to the heat and bring to a boil. Cook, stirring constantly with a wooden spoon until slightly thickened, about 1 minute. Remove from the heat and add the softened cream cheese and salt to the hot mixture. Stir until the cream cheese is blended into the milk and cornstarch mixture.
Pour the hot ice cream base into the bowl with the cream and vanilla. Stir until well blended. Continue to stir every few minutes until the mixture is cooled. Remove the bowl from the ice bath and dry the bottom of the bowl. Cover the bowl and refrigerate for 4-8 hours.
Process the ice cream according to your ice cream maker directions.
Once the ice cream is frozen, spoon ⅓ of the vanilla ice cream in an air-tight freezer safe container. Spoon ⅓ of the strawberry sauce on top and sprinkle with ⅓¼ of the crushed cookies. Repeat twice more with a layer of ice cream, then strawberry sauce, and finally the cookies. Freeze the ice cream at least 4 hours or overnight before serving.
Serve with additional cookies if desired.