Microwave cream and chocolate in a 1-quart glass measuring cup on high power until very hot, about 2 minutes. Whisk until chocolate melts into a smooth sauce.
Line an 8-inch square metal pan with plastic wrap, leaving an overhang. Arrange 1/3 of the graham crackers over bottom, cutting the final few with a knife to fit. Drizzle 1/3 of the sauce over the crackers; spread evenly. Freeze until chocolate is just set, 5 to 10 minutes.
Spread vanilla ice cream over the chocolate. Top with another cracker layer as before. Drizzle another 1/3 of the sauce over the crackers; spread evenly. Freeze until just set, 5 to 10 minutes. Spread chocolate ice cream over the crackers. Arrange a final layer of crackers over the ice cream. Spread remaining sauce over crackers.
Freeze until set, about 2 hours, or double-wrap in plastic and freeze up to 5 days.
To serve, lift dessert from pan; discard plastic wrap. Halve the dessert; cut each half into 6 slices. Return to freezer briefly. Toast marshmallows over a flame and serve alongside s'mores.