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You will definitely need to make the ice cream ahead of time for this. You need to toast the marshmallows before we make the ice cream base. I did this by using my grill, stacking the marshmallows on the end of a two tonged fork and getting them nicely charred. When they are done, you are ready to make the ice cream!
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In a sauce pot over medium low heat add the half and half and cream. Once this is warm add the marshmallows and melt them in the mixture. Stir this occasionally.
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While you are waiting on your marshmallows to melt in a separate bowl add the rest of the ingredients and mix with a whisk until combined and set aside. You want the mixture on the stove to just come to a simmer. Take this off the heat and add about 1/4 cup at a time to the egg mixture stirring constantly until most of the cream mix is added.
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Pour this all back in the pot and heat on medium low heat until thickened. You know this is done when it will coat the back of a spoon. Let the pot sit on the counter for 20 minutes.
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Strain the liquid with a fine mesh colander and pour into an air tight container. Let the mix sit over night in the fridge.
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Churn mixture the next day to the directions on your ice cream maker. Place into another air tight container and let freeze completely in the freezer for at least two hours before assembly.