Heat olive oil in a large pot over medium heat; cook chicken in hot oil until browned, about 5 minutes per side. Add red wine, tarragon, chervil, chives, and parsley to chicken; mix well.
Mix green onions and mushrooms into chicken mixture; simmer, stirring occasionally, until liquid is reduced to a thick sauce and chicken is longer pink in the center, about 1 hour. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
