Place wild rice in a large saucepan. Cover with 2 1/2 cup of water. Bring the water to a boil, the reduce the heat to medium-low, cover and simmer for 30-45 minutes. Drain off excess water and set aside.
In the meantime, preheat oven to 350 degrees. Remove the stems and seeds from the Polano peppers. Place them on a greased cookie sheet and roast for 30 minutes, turning halfway through, until the outsides start to become charred in places. Allow peppers to become cool enough to touch, then remove and discard the thin outer skin. Chop the peppers into smaller pieces.
Heat the oil in a large stock pot over medium heat. Add carrots, celery, and onion. Sauté until onion is translucent, about 5-6 minutes.
Add garlic, thyme, pepper and cumin. Cook until garlic is fragrant, about 30 seconds.
Add flour to pot and stir until flour is incorporated. Cook mixture for about three minutes, stirring constantly.
Add the chicken broth. Bring to a boil, then reduce heat to a simmer and cook until mixture begins to thicken, about 12 minutes.
Stir in whole milk, cayenne pepper, shredded chicken, wild rice and roasted peppers. Cook until mixture thickens again, about 5-10 minutes.
Season with salt and pepper to taste.