Make the Ice Cream Base: 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 (keep a close eye so they don't start smoking!). 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 medium saucepan. Split the vanilla bean down the middle and scrape the seeds into the pan with the milk, then add the vanilla bean to the milk as well. Set the pan over medium heat and bring the mixture to a simmer.
Meanwhile, 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, using a rubber spatula to scrape the mixture from the bowl back into the pan. Turn the heat down to medium-low and cook, stirring constantly with the spatula, until the mixture thickens enough to coat the back of a wooden spoon (it should be between 170 and 175 degrees F on an instant-read thermometer). Strain the custard through a fine-mesh sieve into a heatproof bowl.
Let the mixture cool for about 15 minutes, stirring it every few minutes to help it cool down faster. Put 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 with a lid or plastic wrap and refrigerate overnight.
Make the Fudge Swirl: Whisk together the sugar, water, corn syrup and cocoa powder in a medium saucepan over medium heat, whisking constantly, until the mixture begins to bubble at the edges. Continue to whisk until it just comes to a low boil. Cook for 1 minute, whisking frequently. 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.
Churn the Ice Cream: Before churning, whisk the custard vigorously until it is smooth and has a pourable consistency (it will be a little spongy when it first comes out of the fridge). Freeze in your ice cream maker according to the manufacturer's directions. When it's done churning, gently fold in the chopped graham crackers with a rubber spatula. 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.