At least a day ahead of time: Put your ice cream maker bowl in the freezer!
Cover a sheet pan with a piece of parchment paper. Place 1 cup of marshmallows on the pan.
Put the tray of marshmallows under your oven broiler, and toast until golden brown (about 3 minutes) WATCH CAREFULLY! DO NOT BURN THEM! They will turn from golden to black in the blink of an eye. Set aside and let them cool.
In a large sauce pan/pot, pour in the cream, milk, vanilla extract, and salt and set over medium-low heat. Bring the mixture to a simmer until you see small bubbles around the edges of the pot. Stir occasionally. This will take about 15 to 18 minutes.
In a clean, medium-sized bowl, beat the egg yolks and sugar together with a whisk until the mixture is thick and pale.
Once the cream mixture is simmering, use a ladle to spoon the hot cream mixture into the egg and sugar mixture, adding one ladle of hot liquid at a time. While you are adding the hot cream to the yolks, make sure you are whisking constantly and make sure you are adding the cream SLOWLY. We want to bring the eggs to temperature without allowing them to scramble. Continue until all the hot liquid is added to the yolks.
Transfer the hot cream and egg mixture from the bowl back to the pot, and place it over low heat, stirring constantly. Add in the toasted marshmallows and the other 1 cup of un-toasted marshmallows. Stir until all of the marshmallows dissolve and the custard thickens ~8 minutes.
Once thick, somewhere between the consistency of eggnog and pudding, remove from heat, and pour it through a fine mesh strainer into a large bowl.
Cover the bowl with plastic wrap or aluminum foil and place in the fridge for at least 4 hours, but preferably overnight.
Once the custard has been in the fridge for at least 4 hours, it is now time to churn your ice cream.
Pour the chilled custard in the bowl of your ice cream maker and churn according to your machine's directions.
After about 10-15 min, the ice cream should be thicker. When ready, it will be the consistency of soft serve ice cream, maybe a little looser.
Transfer ice cream to a freezer-safe container and freeze until solid.