In a large bowl, whisk eggs until combined. Whisk in milk, cream, 1/2 cup sugar, vanilla bean, salt, and rum. Add bread and let soak 45 minutes to 1 hour, pushing bread down occasionally until most of liquid has been absorbed.
Meanwhile, in a medium saucepan over high heat, stir together remaining 1 cup sugar, vinegar, and 1/2 cup water until sugar has dissolved. Cook, without stirring, until sugar is dark amber in color, 7 to 10 minutes. Remove from heat and carefully add 1/4 cup water (mixture will sputter); swirl pot until combined.
Working quickly, pour caramel into a 9-inch round baking pan, carefully rotating the pan so caramel covers the bottom and some of the sides; set aside until cool.
Preheat the oven to 350 degrees. Put a kettle of water on to boil. Rub cooled pan with butter and spoon in bread mixture packing bread tightly to fit. Cover with a parchment round.
Place pan in a roasting pan and pour hot water to come halfway up the side of the pan. Bake 1 1/4 hours or until a knife inserted in the center comes out barely clean. Do not overbake. Let stand 5 minutes then run a metal spatula around the edges and invert the pudding onto a large rimmed plate.
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