Grease a 9-inch (23 cm) square baking dish with the butter. Arrange the diced pumpkin bread evenly in the dish and set aside.
In a large bowl, whisk the eggs, salt, sugar, and vanilla until well incorporated. Set aside.
In a small saucepot over medium heat, bring the milk to a heavy simmer. Once simmering, slowly whisk the hot milk into the egg mixture. Note: Whisk in slowly; if you add the hot milk all at once, the eggs will curdle. Slowly pour over the pumpkin bread. Make sure all the bread is soaked with the mixture. Sprinkle the chocolate chips evenly over the top. Transfer to the refrigerator for about 2 hours.
When ready to bake, preheat the oven to 350°F (175°C).
Prepare a water bath by setting the baking dish into a larger ovenproof pan and pouring hot water into the ovenproof pan to come halfway up the sides of the baking dish. Carefully transfer to the oven and bake the pudding for 45 to 50 minutes, or until cooked through. Remove from the oven and serve warm.