In a heavy-bottomed dutch oven, heat the oil over medium-low until it begins to glisten. Add the onions and cook until soft and translucent, about 5 minutes.
Add the poblano peppers and garlic and cook for another minute until fragrant. Add the cumin and chili powder. It'll be pasty but cook for another minute to help bloom the spices. Slowly add the tomato sauce and stir until combined.
Add the broth, beans (undrained), brown sugar, salt, and tarragon and bring up to a simmer.
Add the crushed tortillas and continue to simmer for another 30 minutes to develop the flavors. If you plan to serve the soup later, cover the pot and turn off the heat until you are ready to continue to the next step. About 30 minutes before you plan to serve, use an immersion blender to puree the soup until smooth. For a more velvety texture, put the blended ingredients through a sieve and discard any solids that remain. Place the pot back on the stove and reheat over medium heat. Add the chicken and heavy cream and cook until heated through. While the soup is heating, cut the remaining tortillas into strips and fry in a shallow pan with about 4 T of oil. Drain on a paper towel and sprinkle with salt while still hot. Set aside. Ladle into bowls and garnish with sour cream, cheddar cheese, fried tortilla strips, and cilantro.