Spread sugar evenly in large nonstick skillet. Cook over medium-high heat without stirring until sugar begins to melt and caramelize on bottom (sugar layer will slide when skillet is tilted), then stir until all sugar is melted and deep amber and caramel is smooth, 3 to 4 minutes.

Add 1 cup cream, rum, and ¼ teaspoon salt (some clumps will form). Stir until caramel is thick and smooth, about 1 minute.

Mix in remaining ¼ cup cream.

Remove from heat and add butter; stir until sauce is thick and smooth, about 1 minute. Remove to a bowl or 1 quart measuring bowl to cool and reserve the caramel pan for the bananas.

Preheat oven to 350° F. Spread bread cubes out on rimmed baking sheet. Toast until beginning to feel dry but not hard, about 7 minutes. Cool on sheet.

Whisk eggs, ¾ cup sugar, and ⅛ teaspoon salt in large bowl to blend.

Bring cream and half and half to simmer in medium saucepan. Very gradually whisk cream mixture into egg mixture.

Pour custard into a medium bowl and mix in vanilla. Cool to lukewarm (30 minutes).

Peel the bananas and cut lengthwise in half. Then cut into small pieces.

In the same pan you made the caramel sauce, whisk brown sugar and butter over medium-high heat until smooth sauce forms (1 minute). Whisk in rum, lemon juice, and ¼ teaspoon salt.

Add bananas and toss just until coated, about 30 seconds. Remove from heat.

Lightly butter 13x9x2-inch baking pan. Scatter ½ of bread in dish.

Top with half of bananas and half of syrup from skillet.

Scatter the remaining bread over.

Top with remaining bananas and syrup from skillet.

Pour custard over. Press down on the bread from time to time to ensure the custard is fully absorbed while you preheat oven to 350° F.

Cover the bread pudding with foil. Bake pudding 25 minutes.

Uncover and bake until puffed and set in center, about 20 minutes longer.

Top with caramel sauce.

Serve warm or at room temperature on its own, or with a dallop of crème fraîche, whipped cream, or ice cream. I personally like the tartness of crème fraîche.