Heat the oven to 450°F. Toss corn with oil on a rimmed sheet pan and arrange poblano halves cut-side down beside it. Roast for 20 to 25 minutes, stirring corn once, until corn browns in spots and pepper skin blisters.
Cover the pepper in a bowl for 5 minutes, peel away loose skin and chop the flesh. Transfer roasted corn and poblano to a 6-quart slow cooker.
Add onion, garlic, beans, broth, chili powder, cumin and salt. Cover and cook on low for 4 hours until onion is tender.
Blend about one-third of the soup briefly with an immersion blender, leaving plenty of whole corn and beans. Cover and heat on high for 15 minutes if needed until steaming.
Switch off the heat and stir in sour cream and lime juice. Divide among 6 bowls and top each with one-sixth of the cotija.