Heat olive oil in a large Dutch oven over medium heat. Add onion, poblano peppers, and jalapeño. Cook for 5 minutes until softened.
Stir in garlic, cumin, smoked paprika, chili powder, salt, and pepper. Cook for 1 minute until fragrant.
Add chicken and 4 cups chicken broth. Bring to a gentle boil, reduce heat, cover, and simmer for 15–20 minutes, or until chicken reaches an internal temperature of 165°F.
Remove chicken from the pot and shred using two forks.
Reduce heat to low. Add softened cream cheese and stir until smooth.
Add white beans, salsa verde, and shredded chicken. Simmer for 5–10 minutes.
Gradually stir in cheddar cheese until melted. Add more broth if needed.
Stir in cilantro and fresh lime juice. Adjust seasonings to taste.
Serve warm with avocado, extra cheddar, yogurt, tortilla chips, and fresh lime.