To roast the peppers, heat the oven to 425°F. Line a baking sheet with foil and place peppers on top. Roast for about 30 minutes, turning every 10 minutes, until skins are blistered and pulling away. Place in a bowl and cover to steam. When cool enough to handle, gently peel away skins. Remove stems and seeds and chop peppers.
In a large pot, melt butter over medium heat. Add the onion, carrot, and celery with a pinch of salt and cook 5 minutes. Add the garlic and cook 1 minute more. Stir in the flour to coat the vegetables and soak up any extra butter, then mix in spices and cook 1 minute.
Stir in a small amount of broth to deglaze the pan and scrape up any stuck bits. Add the chicken, corn, poblano peppers, and remaining broth and stir well.
Raise heat to medium-high and bring to a boil, then reduce heat to medium-low or low and let simmer for 10 minutes.
Stir in the cream and simmer 5-10 minutes more, until heated through.
Notes
If you want a heartier soup, use only 6 cups broth. You can also omit the cream if desired.
Use olive oil in place of the butter and omit the cream for a dairy-free version.