In a large soup pot, melt the butter over medium-high heat. Add the onion, celery, carrots, garlic, and poblanos and sauté, stirring often, until tender, 12 to 15 minutes. Add the salt, pepper, cumin, and thyme and sauté until caramelized and fragrant, 3 to 5 minutes longer.
Add the broth and cream, bring to a simmer, then reduce the heat to medium-low heat and cook, stirring often, for 15 to 20 minutes to meld the flavors.
Use an immersion blender to carefully blend the soup until smooth. (Alternatively, let cool slightly and, working in batches as necessary, process in a stand blender until smooth, filling the blender until smooth, filling the blender no more than half full and removing the lid slowly after blending. Pour the soup back into the pot.)
Add the chicken and simmer for 15 to 30 minutes to meld the flavors to your liking. Stir in the cilantro.
Serve warm, garnished with tortilla strips and sliced radishes.
Store in an airtight container in the refrigerator for up to 3 days or in the freezer for up to 1 month.
