Preheat oven to 350 °F and grease a 9-inch square pan. Line pan with parchment paper so that paper hangs over two sides of the pan. Line bottom of pan with a single layer of whole graham crackers (about 4 ½ crackers).
In a saucepot, melt butter and stir in brown sugar, milk and vanilla until dissolved. Remove from heat and stir in graham cracker crumbs (this will make a soft paste). Spread half of the graham mixture over the cracker-lined pan. Sprinkle with a third of the chocolate chunks and half of the marshmallows. Top marshmallows with another layer of whole graham crackers and spread remaining graham mixture over top. Top with another third of chocolate chunks and the remaining marshmallows. Finish with a last sprinkle of chocolate chunks and bake for 12 minutes, until marshmallows have browned. Cool completely before slicing.
Preheat oven to 350 °F and place cut S’mores bars on a parchment-lined baking sheet. Warm squares for 10 minutes, to melt chocolate chunks and soften marshmallows.
Scoop spoonfuls of checkerboard ice cream into bowls and arrange warm, gooey squares on top.
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