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1
Grease 13- by 9-inch baking dish with 1 tablespoon butter. Adjust oven racks to middle and lower middle positions and preheat oven to 325u00b0F.
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2
Arrange bread cubes in single layer on two rimmed baking sheet and bake until dry, 12 to 15 minutes. Transfer sheets to cooling racks and allow to cool completely, about 15 minutes.
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3
Meanwhile, in large bowl whisk egg yolks, eggs, dark brown sugar, 2 teaspoons cinnamon, milk, dulce de leche, rum, vanilla, and salt until thoroughly combined. Add bread and gently mix with rubber spatula until cubes are saturated.
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4
Transfer 1/2 of bread cubes to prepared baking dish. Peel and slice bananas crosswise into 1/2-inch thick slices. Top bread layer half of bananas. Repeat bread-banana layering, finishing with bananas. Melt remaining 3 tablespoons butter and brush on bananas. Combine sugar and remaining 1 teaspoon cinnamon and sprinkle evenly over bananas.
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5
Let mixture stand at room temperature 20 minutes.
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6
Place dish on rimmed baking sheet and bake on middle rack until bread pudding has set and when pressed releases no liquid, about 45 minutes. Transfer pan to cooling rack and cool about 20 minutes before serving.
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7
While bread pudding cools, bring sugar, dark brown sugar, water, rum, cinnamon stick, cloves, and salt to boil in medium saucepan over medium-high heat. Reduce heat to medium and simmer until thickened and the consistency of maple syrup, about 10 minutes. Serve with pudding.
