Preheat oven to 325 degrees. Grease a 2-qt square baking dish; set aside. Place bun pieces on shallow baking sheet. Bake 7-8 minutes or until dry and crisp; cool.
In medium bowl lightly beat eggs. Stir in sweetened condensed milk, 3/4 cup milk, vanilla, and nutmeg; set aside.
Place bun pieces in prepared baking dish. Sprinkle with 1/2 cup each of marshmallows and chocolate chips. Evenly pour milk mixture over all. Let stand 5 minutes Sprinkle with crushed graham crackers. Bake, uncovered, for 35 minutes Sprinkle with 1/4 cup of remaining marshmallows. Bake about 5 minutes more or until knife inserted near center comes out clean.
For drizzle, in a small saucepan heat and whisk remaining marshmallows, remaining chocolate chips, and 2 tbsp milk over low heat until melted and smooth. Drizzle over bread pudding. Cool 20 to 30 minutes before serving. Serve warm.