Heat the broiler with a rack about 6 inches below the element. Arrange whole poblanos on a rimmed baking sheet and broil, turning with tongs, until blistered in patches, about 10–15 minutes. Watch closely. Cover in a heatproof bowl for 10 minutes. Meanwhile, begin the soup base.
Melt butter in a large heavy pot over medium heat. Soften the onion for 6–8 minutes. Add garlic, oregano, and cumin; stir for 30 seconds. Add flour and stir for 1 minute. Gradually whisk in 4 cups broth, then add 1 cup corn. Bring to a gentle simmer and cook for 8 minutes.
When cool enough to handle, peel the loosened skins from the peppers and remove stems, seeds, and pale ribs. Dice 3 peppers. Roughly chop the remaining pepper. Do not rinse the roasted flesh; remove stray skin with your fingers.
Blend the roughly chopped pepper with the remaining 1 cup corn and reserved 1 cup broth until smooth. This mixture is unheated; there is no need to put hot soup in the blender.
Add the blended mixture, diced poblanos, cooked chicken, salt, and black pepper to the pot. Simmer gently for 5 minutes, until the chicken is thoroughly hot. Lower the heat, stir in cream, and warm for 2–3 minutes without a hard boil.
Remove from the heat. Stir in lime juice and taste for salt. If needed, loosen with a little additional broth. Ladle into bowls and scatter cilantro over the top. Serve lime wedges alongside.