Preheat broiler. Line a baking sheet with parchment paper and butter it well. Spread marshmallows in a single layer on the pan. Slide pan under the broiler and toast for about 2-3 minutes, or until tops are golden brown.
Remove from the broiler and flip marshmallows with a buttered spatula. Return to the broiler for another minute to brown the other sides. Set aside to cool for a few minutes.
Transfer marshmallows to a blender. Add cream, half and half, and sugar. Blend on high speed until smooth. Transfer to the refrigerator to chill for an hour.
Set up your ice cream maker. Remove ice cream base from the fridge and whisk it until smooth (it may seize up a bit as it chills). Pour into the machine and process according to your instructions. (If you have a KA ice cream maker, it will take 15 minutes on the lowest setting.)
While the ice cream blends, make the chocolate fudge. Mix ingredients in a small sauce pan and bring to a boil over medium heat, whisking constantly. Boil until sauce begins to thicken. Remove from heat. Sauce will continue to thicken as it cools.
Once the ice cream has finished processing, spread 1/4 cup of chocolate fudge in the bottom of a 1 quart container. Sprinkle with 1/3 of the graham crackers. Spread 1 third of the ice cream on top.
Repeat layers, ending with ice cream on the top.
Freeze for an hour or two to set before serving.