Preheat your oven’s broiler to high.
Place the poblano peppers on a baking sheet and broil for 10-12 minutes, turning occasionally, until the skins are blistered and blackened.
Transfer the peppers to a bowl and cover with plastic wrap. Let them steam for 10 minutes.
Peel the skins off the peppers, remove the stems and seeds, and chop the peppers into bite-sized pieces.
In a large skillet, heat the olive oil over medium heat. Add the diced onion and cook for 4-5 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Transfer the cooked onion and garlic to the slow cooker.
Add the chopped poblano peppers, chicken pieces, diced tomatoes with green chilies, black beans, pinto beans, chicken broth, ground cumin, dried oregano, smoked paprika, cayenne pepper, salt, and black pepper to the slow cooker.
Stir to combine all ingredients.
Cover and cook on low for 6 hours or on high for 3 hours, until the chicken is cooked through and tender.
About 15 minutes before serving, shred the chicken pieces with two forks in the slow cooker.
Stir in the fresh lime juice and half of the chopped cilantro.
Ladle the soup into bowls and top each serving with shredded Monterey Jack cheese and the remaining cilantro.
Serve hot.