1. Roast poblanos under the broiler, turning until blistered, approximately 8–12 minutes. Steam covered for 10 minutes, then peel, seed, and chop.
2. Heat oil in a large pot over medium heat. Cook onion for 4–5 minutes. Add garlic and cook 30 seconds.
3. Add poblanos, green chiles, cumin, chili powder, paprika, and oregano. Stir for 1 minute.
4. Add tomatoes, broth, and chicken. Bring just to a boil, then reduce to a gentle simmer. Cook until each breast reaches 165°F, approximately 15–25 minutes depending on thickness. Remove cooked chicken to a clean plate.
5. Add rice to the broth. Simmer partially covered for approximately 15–20 minutes, or until tender, stirring occasionally. Add hot broth if needed. Stir in corn during the final 5 minutes.
6. Shred chicken and return to the pot. Add pepper and salt to taste.
7. Reduce heat to low. Stir in cream and warm gently. Add cheese a handful at a time over very low heat or off the burner, stirring until melted.
8. Add cilantro and lime to taste. Adjust consistency with broth and serve with optional toppings.

Storage: Refrigerate promptly in shallow containers and use within 3–4 days. Reheat to 165°F, adding broth as needed.

Little tip: Cook the rice separately when making the stew for later meals. It keeps the broth easier to adjust and prevents the grains from absorbing most of the liquid during storage.