Preheat your oven’s broiler. Place the whole poblano peppers on a baking sheet and broil, turning occasionally, until the skins are blackened and blistered, about 8-10 minutes.
Transfer the roasted peppers to a bowl and cover with plastic wrap. Let them steam for 10 minutes to loosen the skins.
Peel off the charred skins from the peppers, then remove the stems and seeds. Chop the peeled peppers and set aside.
In a large skillet, heat olive oil over medium heat. Add the diced onion and cook until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant. Remove from heat.
Add the diced chicken, roasted poblano peppers, cooked onion and garlic mixture, diced tomatoes with juices, black beans, pinto beans, chicken broth, ground cumin, smoked paprika, dried oregano, chili powder, black pepper, and salt to the slow cooker.
Stir everything together to combine.
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 in the slow cooker using two forks.
Stir in the shredded Monterey Jack cheese until melted and incorporated.
Ladle the soup into bowls and garnish each serving with chopped fresh cilantro and a dollop of sour cream.
Serve hot and enjoy your slow cooker chili relleno soup.