Caramel Topped Bread Pudding

Combine eggs in a large bowl using an electric mixer on medium speed until they become light and fluffy, approximately 2 to 3 minutes. Gradually incorporate 1 1/2 cups sugar into the eggs, along with milk, cream, rum, vanilla extract, and salt until everything is fully incorporated. Gently mix in bread cubes to coat them evenly; allow the egg mixture to absorb into the bread, about 1 hour, stirring occasionally.

Combine the remaining 1 cup sugar with 1/2 cup water and vinegar in a heavy saucepan over high heat; cook without stirring until the mixture turns dark amber, approximately 10 minutes. Remove from heat and let it cool down until no more bubbles form. Gradually add the remaining 1/4 cup water to the caramel sauce, stirring continuously until it's fully incorporated.

Pour the warm caramel sauce into a 9x9-inch baking pan; swirl it around to coat the bottom and sides of the pan. Allow the caramel sauce to set, about 15 minutes. Coat the inside of the baking dish and caramel layer with butter. Arrange the soaked bread cubes and any remaining liquid on top of the caramel layer; gently press them together. Cover the pan with parchment paper.

Preheat your oven to 350 degrees F (175 degrees C). Place the baking pan into a large roasting pan; pour enough boiling water into the roasting pan to reach halfway up the sides of the baking pan.

Bake in the preheated oven until a knife inserted into the middle of the pudding comes out clean, approximately 1 hour and 15 minutes. Let it stand for 5 to 10 minutes. Run a knife around the outside of the pan; invert the pudding and caramel sauce onto a serving platter.