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Place guajillo and ancho chilies on a microwave-safe plate and microwave on high at 15-second intervals until hot, pliable, and toasted-smelling, about 30 seconds total. Combine chicken stock, split chicken breast, and toasted chilies in a medium pot. Bring to a boil, reduce to lowest heat setting, cover, and cook until chicken breast is cooked through (an instant-read thermometer inserted into the center of the breast should register 155 to 160°F) and chilies are tender, about 15 minutes.
Remove chicken breast to a plate and set aside to cool. Transfer chilies to the jar of a blender and add 1 cup of stock. Blend, starting at lowest speed and gradually increasing to high speed, until chilies are completely puréed, about 2 minutes. Add cream cheese and blend until homogeneous, about 1 minute, adding more stock as necessary if the mixture is too thick. Whisk chili/cream cheese mixture back into remaining chicken stock. Whisk in vinegar and set aside.
When chicken is cool enough to handle, shred meat into bite-sized pieces using your fingers or two forks. Set aside. Discard skin and bones.
Heat oil in a large saucepan over medium heat until shimmering. Add onion, season with salt and pepper, and cook, stirring, until softened but not browned, about 5 minutes. Add garlic and cook, stirring, until fragrant, about 30 seconds.
Add corn and cook, stirring, until heated through, about 3 minutes. Add tomatoes, chopped green chilies, black beans, and stock/chili/cream cheese mixture. Bring to a simmer over medium heat, stirring frequently. Stir in pepper jack cheese, shredded chicken, and lime juice. Season to taste with more salt and pepper. Stir in cilantro. Serve immediately with diced avocado, picked fresh cilantro leaves, sliced scallions, and grated cheese.