Put the heavy cream, 1 cup of the brown sugar, sugar, rum, eggs, melted butter, kosher salt, and vanilla extract in a large bowl. Whisk until evenly combined.

Cut or tear the brioche bread into smaller 1 inch pieces. Add those to the bowl with the heavy cream mixture and stir until combined. Add a splash or two of the Coke as well.

Let the bread mixture sit for 40 to 45 minutes stirring a few times and pressing so the bread absorbs the liquid.

Place the bread pudding in a well greased 9 by 13 casserole dish and into a 350 degree oven for about one hour, or until the center has set.

Pour the rest of the coke into a saucepan along with the remaining 2 tablespoons of brown sugar. Carefully bring to a boil watching closely so it does not bubble up, if it does bubble simply remove from the heat for a moment. Reduce the heat and let simmer 25 to 30 minutes, stirring a few times, or until it has reduced down.

Note: The Coke reduction will thicken slightly but should still be thin enough to absorb into the bread pudding.

Let the bread pudding cool slightly before serving. Serve with the Coke sauce drizzled over top and whipped cream if desired. Offer more of the Coke sauce on the side so everyone can decide how sweet they like their pudding.