In a medium bowl, combine the chocolate, cream, and corn syrup and microwave for about 1 minute, stirring halfway through and after cooking time, until melted and smooth. Let cool to room temperature, about 30 minutes.
Preheat oven to 325°F/160°C. Spray a 9-inch springform pan with nonstick cooking spray and line the sides with 2 1/2-inch-wide strip of parchment paper. In a medium bowl, combine the graham crackers crumbs, butter, and sugar. Use the bottom of a measuring cup to press the crumbs into an even layer on the bottom of the prepared pan. Bake for about 12 minutes, until fragrant and beginning to brown. Let cool completely.
Pour the cooled chocolate mixture over the cooled crust and smooth into and even layer. Freeze until firm, about 30 minutes.
Spread marshmallow creme over the chocolate in an even layer. Freeze until firm, about 15 minutes.
Spread softened chocolate ice cream evenly over the marshmallow layer. Cover with plastic wrap and freeze until firm, at least 4 hours.
Adjust oven rack to be about 6 inches from broiler element and heat the broiler. Discard the plastic wrap, place the cake on a baking sheet, and arrange marshmallow halves, cut side down, in a layer on top of the ice cream. Broil until marshmallows are lightly browned, 30-60 seconds, rotating halfway through. Refreeze cake, if necessary.
Working quickly, remove pan ring and parchment paper and place graham crackers quarters vertically along the sides of the cake.