Roast the poblano peppers over an open flame or in the oven until charred. Steam them for 10 minutes, peel, remove seeds, and dice.
If using raw chicken, boil or poach until cooked, about 15-20 minutes, then shred.
In a large pot, heat olive oil over medium heat. Sauté onion for 5 minutes, add garlic and cook for 1 minute.
Add diced poblano peppers, cumin, smoked paprika, salt, and black pepper. Cook for 2-3 minutes.
Pour in chicken broth, bring to a boil, then simmer for 10 minutes. Blend if desired.
Add shredded chicken and corn, simmer for 5 minutes. Stir in heavy cream, cilantro, and lime juice.