Cut bread into ½ inch cubes and spread it out on a large baking sheet. Leave it out to get stale for about a day.
Preheat the oven to 375 and lightly grease a 9x13 casserole dish.
Spread stale bread in the baking dish evenly.
In a large bowl, whisk together eggs, sugar, milk, evaporated milk, pumpkin puree, melted butter, vanilla extract, pumpkin spice extract, pumpkin pie spice, and cinnamon. Whisk until all smooth and combined evenly.
Pour pumpkin mixture over the Brioche bread cubes evenly, all over. Press down all over the bread so that it all gets soaked.
Bake bread pudding for 45-50 minutes. (If the top is browning too fast, you can cover the top loosely with a sheet of aluminum foil half way through or so.)
Heat a cooking pan over medium heat and melt butter. Add sugar and cinnamon, stir well. Add spiced rum and simmer for 3-5 minutes, stirring slowly. Take off heat until ready to serve. Pour some sauce over bread pudding slices while serving. (If not using all the sauce at once, you can keep it in the refrigerator, in an air-tight container. Reheat and stir before using again.)