Heat oil in a large skillet over medium heat. Add chicken and brown, 4 to 5 minutes per side. Remove chicken to a plate.
Add mushrooms, onion, and celery to the pan drippings and sauté until soft, about 5 minutes. Add garlic and cook until fragrant, about 2 more minutes.
Return chicken to the skillet. Add white wine and cook until it has almost completely evaporated, 2 to 3 minutes. Add diced tomatoes, chicken broth, parsley, oregano, pepper flakes, salt, and pepper. Cover and simmer until chicken is no longer pink at the bone and the juices run clear, about 30 minutes.