Preheat the broiler and brush a baking sheet with some vegetable oil. Let the ice cream sit at room temperature until soft. Chill a medium stainless steel bowl in the freezer.
Scatter 2 cups of the marshmallows on the prepared baking sheet and broil, rotating the pan as needed, until very dark and toasted, 3 to 4 minutes. Let cool. Scrape up the marshmallows, transfer them to a cutting board and roughly chop (lightly oil the blade of your knife to help keep the marshmallows from sticking). Transfer the soft ice cream to the frozen bowl, add the chopped marshmallows and stir to combine. Transfer the ice cream to a resealable container and freeze until solid, about 3 hours. The ice cream can be made up to a day ahead.
Put the chocolate, corn syrup, vanilla and a pinch of salt in a medium bowl. Heat the cream in a microwave-safe bowl in the microwave until very hot, about 1 minute. Pour over the chocolate and let sit for 2 minutes. Stir until all the chocolate is melted and smooth. Pour into a small baking dish and refrigerate until set, about 25 minutes (the chocolate mixture should still be soft and scoopable). The chocolate mixture can be made up to a day ahead.
Lay the graham crackers out on a baking sheet in a single layer. Spoon a teaspoon of the chocolate mixture onto each graham cracker and gently spread it out to the edges, being careful not to break the cookies.
Let the ice cream soften slightly at room temperature. Top a graham cracker with about 1/4 cup of the ice cream. Sandwich with a second graham cracker, chocolate-side down, and press some of the remaining untoasted marshmallows into the sides to decorate. Repeat with the remaining graham crackers and immediately freeze until solid, about 1 hour. Once frozen solid, wrap individually and freeze for up to 1 week.
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