Sauté the vegetables. Set the Instant Pot to Sauté mode. Add the olive oil, onion, poblano, and jalapeño and cook, stirring occasionally, until the vegetables begin to soften, about 5–6 minutes. Add the garlic and cook for 30 seconds.
Bloom the spices. Stir in the cumin, oregano, coriander, chili powder, salt, and black pepper. Cook, stirring constantly, for about 30 seconds, until fragrant
Deglaze the pot. Pour in the chicken broth and stir in the Better Than Bouillon. Scrape the bottom of the pot thoroughly with a wooden spoon to loosen any browned bits.
Add the chicken and remaining ingredients. Nestle the whole chicken breasts into the broth. Add the Great Northern beans, diced green chiles, and corn. Do not stir.
Pressure cook. Secure the lid and set the valve to Sealing. Cook on High Pressure for 8 minutes. When the cooking time is complete, allow the pressure to release naturally for 10 minutes, then carefully release any remaining pressure.
Cut the chicken. Carefully open the lid and transfer the chicken breasts to a cutting board. Let them rest for a few minutes, then cut or shred the chicken into bite-size pieces.
Thicken the chili. Transfer about 1 cup of the beans and cooking liquid to a blender and blend until smooth. Stir the puréed mixture back into the chili.
Make it creamy. Add the softened cream cheese to the pot and stir until completely melted and the broth is smooth and creamy.
Finish the chili. Return the chicken to the pot and stir gently to combine. Stir in the fresh lime juice and cilantro. Taste and adjust the salt, pepper, or lime juice as needed.
Serve. Ladle the chili into bowls and serve hot with your favorite toppings.