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Pat the chicken breasts dry and sprinkle them with salt and pepper. Let sit for 10-15 minutes.
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Heat the avocado oil in a cooktop-safe slow cooker insert or a large pan. When it is shimmering, add the chicken. Brown on each side, about 6 minutes per side. Remove to a bowl.
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Add a tiny bit more oil if needed at any point in the browning stages.
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Add the potatoes with a pinch of salt--do not overcrowd the pan, you want them to brown, not steam. Brown the potatoes on a few sides, in batches if necessary. When they are brown, remove them to the bowl with the chicken.
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Brown the mushrooms in batches similarly, with a pinch of salt per batch. Remove to the bowl with the chicken and potatoes.
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Add the onion with a pinch of salt. Cook, stirring, for 5 minutes. Add 1-2 tablespoons of the vinegar, the larger amount if there is a lot sticking to the bottom of the pan, less if not. Add the roasted peppers and garlic to the pan. Cook for 1-2 minutes. Add the cumin and stir. Cook for 1 minute.
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If your slow cooker insert is not cooktop safe, transfer the ingredients in the pan to the insert now.
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Add the chicken stock, the roasted chicken, potatoes and mushrooms, along with all accumulated juices, and the corn cob. Transfer to the slow cooker base and cook on low for 2-3 hours.
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An hour before serving, add the corn kernels and diced zucchini and turn the setting to high. Cook for 1 hour on high.
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Remove the chicken and shred the meat. Return to the soup.
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Right before serving, mix in the cream and cream cheese. If it tastes dull, mix in a tablespoon of apple cider vinegar. Taste for more salt.
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Serve with sour cream, shredded cheese, and, if it still tastes a little dull, tomato salsa.