Preheat the oven broiler to high.
Place the poblano peppers on a baking sheet and broil for 8 to 10 minutes, turning occasionally, until the skins are charred and blistered.
Transfer the peppers to a bowl and cover tightly with plastic wrap. Let them steam for 10 minutes.
Peel the skins off the peppers, remove the seeds and stems, and chop the peppers into bite-sized pieces.
Heat the olive oil in a skillet over medium heat. Add the diced onion and cook for 3 to 4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant. Remove from heat.
In the slow cooker, combine the chopped poblano peppers, cooked onion and garlic, chicken pieces, diced tomatoes with green chilies, black beans, white beans, chicken broth, ground cumin, dried oregano, smoked paprika, chili powder, black pepper, and salt.
Stir well 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 and stir in the chopped fresh cilantro.
Ladle the soup into bowls and top each serving with shredded Monterey Jack cheese and a dollop of sour cream.
Garnish with additional cilantro if desired and serve hot.