1. Broil the poblanos, turning, until blistered, about 8–12 minutes. Cover for 10 minutes; peel, seed, and chop. Brown the corn in a dry skillet for about 5–7 minutes if not already roasted.
2. Heat the oil in a large pot. Cook the onion for 4–5 minutes. Stir in the garlic, cumin, chili powder, paprika, oregano, and pepper for 30 seconds.
3. Add the poblanos, corn, enchilada sauce, tomatoes, beans, and broth. Bring to a gentle simmer.
4. Add the chicken, partially cover, and simmer about 20–25 minutes, or until the thickest part reaches 165°F. Remove to a clean plate, shred, and return to the pot.
5. Reduce heat to low. Stir in cream cheese until melted, then add the heavy cream and warm gently for 3–5 minutes. Add Monterey Jack by handfuls, stirring until melted. Do not boil.
6. Add lime juice and salt to taste. Thin with broth if desired. Serve with toppings.

Storage: Refrigerate in shallow airtight containers within 2 hours and use within 3–4 days. Reheat gently to 165°F. For best freezing texture, freeze the soup base before adding dairy for up to 3 months, then thaw in the refrigerator and finish with dairy and lime.