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To make the pudding:
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Heat the oven to 325F.
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Grease a 13 x 9-inch baking dish with the softened butter and set aside.
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Place the bread cubes on a rimmed baking sheet and toast in the oven until golden-brown, 12 to 15 minutes, rotating midway through.
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Set aside to cool.
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Place the bread in the prepared baking dish and set aside.
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Whisk the eggs in a medium bowl and set aside.
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Sift the cocoa into a medium bowl and set aside.
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Place the chocolate in a large bowl.
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Bring the milk, cream, sugar, and salt to a boil in a large pot, stirring occasionally to dissolve the sugar.
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Turn off the heat, stir in the vanilla, and then pour the hot mixture over the chopped chocolate.
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Cover the bowl with plastic wrap, set aside for 5 minutes, and then whisk until smooth.
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Whisk 1/2 cup of the chocolate mixture into the cocoa, stirring until smooth.
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Whisk in another 1/2 cup of the chocolate mixture and then whisk in the eggs.
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Transfer to the large bowl of remaining chocolate mixture and whisk until they are completely incorporated.
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Pour all but 1 cup of the chocolate mixture over the bread cubes in the baking dish and set aside so the bread can soak up the liquid.
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Press down on the bread with a wooden spoon every 15 minutes for 1 hour, adding the rest of the chocolate mixture after about 30 minutes, or when the bread has soaked up enough so the last cup of liquid will fit.
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Heat the oven to 350F.
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Cover the bread pudding with aluminum foil and use a paring knife to make 4 small slits in the foil to allow steam to escape.
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Set the baking dish in a large roasting pan and place in the oven.
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Pour enough hot water in the roasting pan so the water reaches 1 inch up the side of the baking dish (if you dont have a roasting pan large enough to fit the baking dish, set the dish onto a rimmed baking sheet and slide it in the oven, adding enough water to the baking sheet so it cushions the baking dish but doesnt spill over).
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Bake for 45 minutes, and then remove the foil and bake until the pudding begins to puff slightly and the center bounces back to light pressure, about another 25 to 35 minutes.
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Cool for 30 minutes.
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To make the sauce:
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While the bread pudding cools, make the caramel.
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Place the sugar in a 2-quart saucepan and add 1/4 cup of water.
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Cover (or if you cant find a lid, top the saucepan with a heatproof bowl, making sure the bottom of the bowl doesnt touch the sugar) and cook over medium heat, swirling the mixture every 1 to 2 minutes, until the sugar is liquefied, about 6 minutes.
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Continue to cook until the sugar is a medium-amber color, another 4 to 6 minutes.
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Turn off the heat and add the cream (it will vigorously bubble up at first), whisking the mixture until smooth, then add the softened butter, bourbon, and salt.
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Set aside and serve with the still-warm bread pudding.
