Peel and dice 3 ripe mangoes and transfer to a casserole dish.
In a small bowl mix ½ cup sugarand 2 teaspoons cornstarch together until blended and pour it over the mangoes. Stir to combine and transfer to the oven. Cook the fruit for 20 minutes. Remove and cool to room temperature.
Transfer about 1 cup of the mango mixture to a mini food processor and pulse to puree. Set aside. (This can be made several days ahead).
FOR THE MANGO ICE CREAM:
Reserve 2 tablespoons of the 2⅔ cups whole milk in a separate prep bowl and whisk in 5 teaspoons cornstarch. Set aside.
In a medium bowl, beat ¼ cup cream cheese and¼ teaspoon fine sea salt or kosher salt with a hand mixer until smooth and evenly blended. Set aside.
In a large 4=quart saucepan, heat the remainder of the milk with 1½ cups cream, ⅔ cup granulated sugar, ¼ cup light corn syrup. Bring to a boil, stirring constantly and cook for 4 minutes.
Add the milk and cornstarch mixture to the cream mixture and cook, stirring constantly until it thickens.
Remove the cream mixture from the heat and add ¼-⅓ cup of the hot cream to the cream cheese. Use the hand mixer to beat until smooth, then gradually, beat in the remainder of the cream mixture until the base is well blended.
Add ⅔ cup roasted mango puree to the custard and stir to blend. (reserve leftover mango to swirl as a mix-in when packing the dessert.)
Place a mesh strainer over a large bowl and pour the custard through the strainer to remove any stringy pulp. Stir in ¼ cup golden rum. Cover the custard with plastic wrap and refrigerate for several hours until very cold.
CHURN:
Churn the custard according to the manufacturer's instructions on the machine. It should take 20-25 minutes to become like soft-serve.
Pack the frozen dessert, alternating with the leftover roasted fruit, into a container and smooth the top. Cover and freeze for 2-4 hours until solid, then scoop and enjoy.
Video
Notes
Note: to swirl the leftover fruit puree, add a cup of ice cream to the container, followed by a spoonful of mango puree. Continue to alternate in this way until it's all used. Freeze well (preferably overnight) before scooping.