In a small bowl, mix ¼ cup whole milk with cornstarch to create a slurry. Set aside.
In a heavy 4 quart saucepan, combine remaining milk, cream, sugar, and salt. Bring to a boil over medium heat, stirring occasionally, and allow to boil for 4 minutes. Watch carefully and adjust heat as necessary to avoid boil over.
Remove from the heat and stir in the slurry.
Return to the heat and bring to a boil once again. Cook, stirring, until thickened, about 2 minutes. Remove from the heat and stir in vanilla.
Place the cream cheese in a large bowl and slowly whisk in the hot milk mixture. Continue to whisk until completely incorporated and smooth.
Transfer the mixture to a one gallon resealable plastic bag and seal. Submerge the bag in a bowl of ice water, kneading the bag every ten minutes or so and adding more ice as necessary, until chilled, about 30 minutes.
While the base is chilling, combine butter and brown sugar in a medium sized bowl.
Add the molasses and thoroughly combine.
Add flour, ginger, cinnamon, and cloves and stir until almost incorporated.
Add the heavy cream and thoroughly combine. Set aside.
Pour the ice cream base mixture into an ice cream maker and freeze according to manufacturer’s instructions.
Pour half the churned ice cream into a one quart storage container with a tight fitting lid, then crumble the cookie dough over the top. Pour in remaining ice cream and smooth the top. Seal the top of the container and place in the coldest part of the freezer until firm, at least 4 hours. Let sit at room temperature for a few minutes before scooping and serving.