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See note about special equipment.
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Put the chicken breasts on a plate large enough to fit the chicken without overlapping. Pat the chicken dry with paper towels. Season the chicken breasts on both sides with Kosher salt, about a half teaspoon in total. Set the plate of chicken aside to rest.
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Adjust the oven rack to be 8 inches below the broiler element. Turn on the Broiler.
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Place all the chili peppers, onions and peeled garlic cloves on a foil lined rimmed baking sheet. Drizzle 1 Tb of vegetable oil and half teaspoon of Kosher salt over the vegetables and toss with your hands until they are evenly coated with the oil and salt.
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Place the baking sheet with the vegetables under the broiler, and broil them until they have browned and the skins are wrinkled. Turn the vegetables from time to time for even browning. This can take a while, about 20 minutes. You want the skin on the peppers to be wrinkled all over as well as browned. Remove any vegetables, like the garlic and onions, if they are done before the peppers. You do not want the garlic to be completely burned.
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Pour the chicken stock into a large bowl and place the broiled chili peppers in the stock. The stock will help the skin separate from the flesh of each pepper in one long piece. Peel off the skin and remove the seeds from each pepper in the stock. As you finish peeling the skin off each pepper, place the peeled pepper into a small bowl.
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In the bowl of a food processor or blender, add the peeled peppers, broiled onions and garlic and puree until smooth. Set aside
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Rise the canned beans under cold running water and set aside.
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Place a large Dutch oven (mine was 5 qt size) on the stove and turn the heat to medium. Add the remaining 2 Tb of vegetable oil and heat until the oil is very hot but not smoking. Add the ground cumin, ground coriander seed, dried oregano, and bay leaf to the shimmering hot oil and stir until fragrant, about 30 seconds. Add the reserved chili pepper and onion puree and stir until incorporated.
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Place a fine mesh strainer over the Dutch oven with the chili puree, and pour the chicken stock through the strainer into the pot. Press down on the chili pepper skins and seeds with the back of a spoon, to push out as much liquid as possible, being careful not to push the skins and seeds through the mesh. Dispose of the skin and seeds. Stir the chicken stock with the chili pepper puree until combined.
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Add the rinsed canned beans (or soaked and rinsed dry beans) and chicken breasts to the stock. Bring the liquid up to a boil, then reduce the heat to a gentle simmer. Occasionally stir the chili and cook the chicken until it is slightly underdone about 15 - 20 minutes. The internal temperature of the chicken should be between 150 degrees F and 155 degrees F. Remove the chicken and place on a plate and let the chicken rest.
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Continue to simmer the chili for 30 minutes if using canned beans, 1 hour or until soft if using dried beans, stirring occasionally. When the beans are very soft, measure out a cup and a half of the cooked beans and their liquid, then add to the bowl of a food processor or blender. Puree the beans until very smooth. Stir the pureed beans back into the pot with the chili.
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Shred the chicken into bite size pieces and place back into the chili. Stir and simmer the chili for about 5 minutes. Taste and add salt as needed, then stir to incorporate. Add fresh squeezed juice from one lime, one handful of the chopped cilantro, and the minced green onions. Stir to combine the added ingredients and taste. Adjust seasoning with salt if needed.
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Cut the remaining lime into small wedges and place in a small serving bowl.
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Ladle a bowl with the chili and garnish the chili with a sprinkle of chopped cilantro, and a couple of slices of pickled jalapenos, and grated Pepper Jack cheese.
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Offer lime wedges, avocado slices, more pickled jalapenos and grated cheese at the table for guests to help themselves.