Melt butter in a large pot over medium-high heat. Add chopped onions, celery, carrots, garlic, and poblanos, saute until tender, about 15 minutes. Add the salt, pepper, cumin, and thyme. Saute 5 minutes longer.
Add the broth and bring to a boil, then reduce the heat to medium-low, add raw chicken. Cover and let simmer for 15 minutes or until chicken is cooked through.
Remove the cooked chicken to a plate and shred. Set aside.
Add heavy cream to the soup pot, stir until blended.
Use an immersion blender to carefully blend the soup until smooth. (You may use a blender instead, but working in small batches. Blend the soup until smooth).
Add the shredded chicken back to the pot. Stir in cilantro. Let simmer for an additional 10 minutes to bring the flavors together.
Serve with tortilla strips, sliced radishes and cilantro.
This will stay fresh in the refrigerator for up to 3 days or you can freeze for 1 month.