Line an 11×7-inch baking dish with parchment paper. Sprinkle 1/2 of the crushed graham crackers on the bottom of the dish. Set aside.
Melt chocolate chips, condensed milk, butter, and vanilla in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching.
Pour 1/2 of the melted chocolate mixture into the prepared baking dish; sprinkle 1/2 of the marshmallows over chocolate mixture. Repeat layers with remaining crushed graham crackers, chocolate mixture, and marshmallows.
Chill in the refrigerator until solid, 3 hours to overnight.