Step 1

Brown chicken in olive oil in a Dutch oven over high heat until nice and dark on both sides and fat has rendered, 6 to 8 minutes per side. Remove chicken to a bowl.

Step 2

Reduce heat to medium and add onion, mushrooms, and salt to the pot. Cook, stirring occasionally, until well browned, 8 to 10 minutes. Add chili powder, cayenne, black pepper, cumin, dried oregano, garlic, and poblano peppers. Cook and stir until peppers begin to soften, about 3 minutes.

Step 3

Add diced tomatoes, chicken broth, and bay leaf; stir and increase heat to high. Stir in 1 cup creme fraiche and fresh oregano and bring to a simmer. Add chicken and any accumulated juices to the pot, nestling them into the broth so they are covered. Reduce heat to low or medium-low to maintain a gentle, steady simmer. Simmer, stirring occasionally, until chicken is cooked through, about 1 hour 15 minutes.

Step 4

Use tongs to carefully transfer chicken to a bowl and let cool enough to handle. Increase heat to medium-high. Cook until braising liquids have reduced by about half, 15 to 20 minutes, skimming off fat from the surface as it cooks.

Step 5

Meanwhile, remove and discard skin and bones from chicken (or save to make stock). Break up the meat into large or small pieces and save any edible drippings.

Step 6

Reduce heat to low and add chicken meat and drippings to the sauce. Simmer until chicken has absorbed the flavor from the sauce, about 15 minutes. Turn off heat and stir in chopped cilantro and remaining creme fraiche. Taste to adjust salt and seasonings.