Preheat oven to 350 degrees. Tear hamburger buns into bite-sized pieces. Toss in a large bowl with the chocolate chips. Pour bread and chocolate chips into a greased 9x13 baking dish.

In a large bowl, whisk eggs and sugar together. Add half-and-half and vanilla and whisk to combine. Pour custard over the bread mixture. If necessary, press down on the bread to make sure it is all covered with custard. Sprinkle additional chocolate chips over top.

Bake 50-60 minutes or until the top is a golden brown color and custard is set. Allow to cool 20 minutes before serving. Serve warm or at room temperature with a big scoop of salted caramel ice cream and a drizzle of chocolate syrup.