Place a medium bowl inside a larger container filled with ice water. This will be used to cool the custard later. Set aside.

Line a baking sheet with parchment paper. Arrange marshmallows in a single layer and place under the oven broiler for approximately 30 seconds to 1 minute, until golden brown. Monitor closely to avoid burning. Using tongs, flip the marshmallows and broil the other side. Once both sides are toasted, set them aside.

In a large saucepan over medium-high heat, combine the milk, heavy cream, and ¼ cup of sugar. Stir occasionally and heat until steam forms and the temperature reaches approximately 175°F.

In a separate bowl, whisk together the egg yolks and remaining ¼ cup of sugar until smooth. Gradually whisk in about 1 cup of the hot cream mixture to temper the yolks, then return the tempered yolk mixture to the saucepan with the remaining hot liquid. Continue cooking over medium heat, stirring constantly, until the custard thickens and reaches a temperature of 180–185°F. Do not allow it to boil.

Once thickened, immediately pour the custard into the prepared bowl over the ice bath. Stir occasionally for approximately 15 minutes until cooled. Remove the bowl from the ice bath and stir in the vanilla extract.

Transfer the toasted marshmallows to a blender. Add the cooled custard and blend for about 2 minutes, or until smooth. Pour the mixture into a clean bowl, cover tightly with plastic wrap, and refrigerate for at least 4 hours or up to 24 hours until thoroughly chilled (about 40°F).

In a medium saucepan over medium heat, whisk together sugar, water, corn syrup, and cocoa powder. Heat until the mixture begins to bubble around the edges. Allow it to come to a gentle boil and cook for 1 minute, whisking frequently. Remove from heat, stir in vanilla, and refrigerate until fully chilled (minimum 2 hours).

Once the custard is chilled, whisk it briefly to restore a smooth, pourable consistency. Pour into your ice cream machine and churn according to the manufacturer’s instructions. When nearly finished, fold in the crushed graham crackers.

Spread a spoonful of the chilled fudge sauce into the bottom of a freezer-safe container. Layer freshly churned ice cream with alternating dollops of fudge swirl. Avoid mixing the layers to maintain a clean swirl effect. Cover and freeze until firm, at least 2 hours.