Place flour in a shallow bowl. Dredge chicken through flour until evenly coated, shaking excess flour back into bowl. Reserve remaining flour.
Heat olive oil in a large skillet over medium-high heat; saute chicken until golden brown, about 3 minutes per side. Add mushrooms, carrots, cooking wine, bacon, chicken broth, garlic, shallot, parsley, thyme, marjoram, bay leaf, salt, and black pepper; bring to a boil.
Reduce heat to medium-low, cover skillet, and simmer until chicken is cooked through and carrots are soft, 15 to 20 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Mix butter and 2 tablespoons reserved flour together in a bowl until smooth; stir into chicken mixture until sauce becomes thick, about 5 minutes.