Heat your broiler. Spread the marshmallows in a single layer on a parchment-lined baking sheet. Place them under the broiler for 30 seconds to 1 minute, or until toasted and golden. Take them out and, using tongs, flip them over and pop them back under the broiler for another 30 seconds or so, until the other side is toasted. (Alternately, you can use a kitchen torch to toast them right there on the pan.) Set the marshmallows aside.

Place the milk in a heavy-bottomed saucepan. Split the vanilla bean down the middle and scrape the seeds into the pan with the milk. Set the pan over medium heat and bring the mixture to a simmer.

While the milk is coming up to a simmer, whisk together the egg yolks, sugar and salt in a large, heatproof bowl. Slowly add the warm milk mixture to the egg mixture, whisking constantly. Pour the mixture back into the saucepan. Put the heat on medium-low and cook, stirring constantly, until the mixture thickens enough to coat the back of a wooden spoon (it should be between 170 and 175 degrees F on a thermometer). Strain through a fine-mesh sieve into a heatproof bowl. Allow to cool for about 20 minutes before proceeding.

Place the toasted marshmallows in the blender, then pour the custard into the blender as well. Puree until the marshmallows are completely broken down and the mixture is smooth. Add the heavy cream and vanilla extract and blend again for about 5 to 10 seconds to combine. Pour the mixture into a bowl, cover andrefrigerate overnight.

Whisk together the sugar, water, corn syrup and cocoa powder in a medium saucepan over medium heat until the mixture begins to bubble at the edges. Cook for about 1 minute, whisking the entire time. Remove from the heat, stir in the vanilla and let cool. Chill in the refrigerator for at least 2 hours before using. It can be stored for up to 2 weeks, covered, in the refrigerator.

Freeze in your ice cream maker according to the manufacturer’s directions. When it’s done churning, gently fold in the chopped graham crackers. Spoon some of the fudge swirl onto the bottom of the storage container. As you remove the ice cream from the machine, layer generous spoonfuls of the sauce between layers of the ice cream, ending with a top layer of ice cream. Do not stir the fudge into the ice cream, as it will make the ice cream muddy looking. Freeze until the ice cream is firm, at least 2 hours.