Mix together milk, whipping cream, sugar, salt and vanilla. Pour into the insert of ice cream maker and make according to ice cream maker instructions. In the meantime, place marshmallows on a baking sheet and broil for about 3 minutes, taking care not to burn. Remove from oven and allow to cool.
Once ice cream is almost ready, add crumbled graham crackers and chocolate pieces. Tear apart marshmallows and add pieces to ice cream as well. Continue making ice cream until it has completely made. Serve immediately for softserve ice cream or place in an airtight container and place in freezer for a few hours or overnight for a firmer ice cream.