Heat the olive oil in a large pot or Dutch oven over medium heat.
Add the onion, poblano or green bell pepper, and jalapeño. Cook for 5 to 6 minutes, stirring occasionally, until softened.
Add the garlic, cumin, chili powder, smoked paprika, coriander, oregano, salt, black pepper, and cayenne pepper. Cook for 30 to 45 seconds, stirring constantly, until fragrant. If the spices begin to stick, add a small splash of chicken broth.
Add the chicken, chicken broth, 2 cans of white beans, diced green chiles, and salsa verde. Stir to combine.
Mash the remaining can of white beans with a fork or potato masher until mostly smooth, then stir the mashed beans into the pot. For an extra creamy chili, blend the beans with 1/2 cup of the broth before adding them.
Bring the chili to a gentle simmer. Reduce the heat to medium-low, cover, and cook for 20 to 25 minutes for chicken breasts or 25 to 30 minutes for chicken thighs, until the chicken reaches 165°F in the center and shreds easily.
Remove the chicken from the pot and shred it with two forks.
Return the shredded chicken to the pot. Stir in the frozen corn and simmer uncovered for 5 minutes. If the chili is thicker than you like, add a splash of chicken broth.
Place the softened cream cheese in a small bowl. Add about 1/2 cup of hot broth from the chili and whisk until smooth.
Reduce the heat to low. Stir the cream cheese mixture into the chili until melted and creamy.
Turn off the heat and let the chili cool for 2 minutes. Stir in the sour cream, Monterey Jack cheese if using, and 1 tablespoon lime juice. Do not boil after adding the dairy.
Taste and adjust with more salt, pepper, lime juice, or cayenne as needed. Add the second tablespoon of lime juice only if you want a brighter, tangier chili.
Serve warm with cilantro, avocado, crushed tortilla chips, green onions, jalapeños, shredded cheese, or extra lime wedges.