The bread pudding recipe is an original from Mulate's Restaurant in New Orleans. It was given to me by my friend Rene. The rum sauce is different from the one at Mulate's. We just couldn't seem to get their sauce "just right."
Preheat oven to 350 degrees. Grease a 9x13-inch pan and set aside.
In a large mixing bowl, whisk the eggs. Add the milk, half and half, sugar, and vanilla. Whisk until combined.
Break the hamburger buns into pieces and place in the prepared pan. Sprinkle the raisins evenly over the buns and pour the liquid mixture over all. Make sure that all of the bread is thoroughly soaked in the liquid.
Bake in the middle rack of the oven for 45 minutes. Let cool for 15 minutes before serving. If you prefer a firmer texture, chill about 2 hours, then reheat for serving. Serve with Caramel Rum Sauce.
CARAMEL RUM SAUCE: Combine sugar and cornstarch in a heavy saucepan. Stir in cream, water, and corn syrup.
Cook and stir over medium heat until thickened and bubbly (mixture may look curdled. Not to worry. Cook and stir for 2 minutes more.
Remove from heat and stir in butter, vanilla, and rum. Let mixture stand at room temperature until cooled. Store in refrigerator for up to 2 months.