Place a large pot over medium heat and add the milk to it. Once it starts to get warm add the marshmallows and stir frequently until the marshmallows have all melted.Leave to cool.
Heat heavy cream on the stove or in the microwave till almost boiling. Meanwhile beat egg yolks and sugar together in a separate bowl. Whisking continuously, drizzle the hot cream into the egg mixture. Mix in the vanilla.
Add the cooled marshmallow mixture to the egg mixture. Whisk till homogeneous.
Churn in ice cream maker according to manufacturer’s directions and transfer ice cream to a container for the freezer. Serve with caramel sauce if desired.