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Put the washed beans, water and salt in the Instant Pot. Seal the top and cook on high pressure for 20 minutes. Let steam release naturally for 20 minutes and then vent the top, using oven mitts and exercising caution not to be burned by the residual steam.
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Drain the beans and wipe out the pot until it is clean and dry.
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While the beans are cooking, broil the poblano peppers until they are blistered and charred all over. Place them in a plastic bag and let them steam and cool down. Remove the skin, seeds and ribs from the peppers and chop them finely.
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Turn Instant Pot to the Saute function and add 2 tablespoons of olive oil. Saute the diced chicken in two batches until it is just slightly brown and all of the pink color is gone from the meat. This will take 8-10 minutes for each batch. The meat should be cooked through but not overcooked. Remove the chicken and set aside.
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In the empty pot which is still on the Saute function, add the remaining oil, onions, salt and peppers. Saute until the onions are soft and translucent, about 10 minutes. Add the cumin and stir. Rub the marjoram and oregano between your fingers to release the flavor and add that to the pot along with the garlic. Stir for 1 minute only. Turn the Saute function off.
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Add the chicken stock to the pot with the onions, garlic, peppers and spices. Add the tied cilantro stems and cook the mixture on HIGH pressure for 30 minutes, allowing the pressure to release naturally. Open the pot and remove the cilantro stems. Remove the pot insert. Stir in the cooked reserved chicken, half of the yellow cheddar and half of the white cheddar until all of it is melted and the chili is smooth.
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Serve the chili in bowls garnished with the chopped cilantro leaves and passing the rest of the shredded cheeses, avocado slices and sour cream.
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The white chicken chili can also be made on the stovetop. Soak the beans overnight, drain and rinse them and cook them in 8 cups of water with 1 teaspoon of salt for 1 ½ hours. Drain and set aside. Saute the chicken, onions, peppers, spices and garlic as above for the instant pot version. Proceed with the recipe on the stovetop, cooking the beans in the stock for another hour. Add the chicken and cheeses as above and serve.