Heat the olive oil in a large skillet over medium-high heat. Add the ground turkey and onion and cook, breaking the meat into crumbles, for 6 to 8 minutes until no pink remains.
Add the garlic, cumin, oregano, chili powder, and smoked paprika. Stir for 45 seconds until fragrant.
Transfer the mixture to a 6-quart slow cooker. Add the poblano, cannellini beans, great northern beans, green chiles, salsa verde, broth, salt, and pepper. Stir to combine.
Cover and cook on LOW for 6 hours or HIGH for 3 hours, until the poblano is tender.
Press about 1 cup of the beans against the side of the slow cooker with a wooden spoon to thicken the chili, then stir.
Add the cubed cream cheese, cover, and let sit for 20 minutes. Whisk until smooth.
Stir in the sour cream, lime juice, and cilantro. Taste and adjust salt. Serve hot with your favorite toppings.