Preheat oven to 350 F. Whisk 1/4 flour and all "chicken seasoning" together in a shallow dish. Pat chicken dry and dredge each piece in the mixture, pressing the flour into the chicken, shake off any excess, then transfer to a dry surface (reserve any leftover flour).
Melt 1 tablespoon butter in 2 tablespoons oil over medium-high heat in a deep, oven-proof skillet/braiser. Once hot, add chicken and cook approximately 4 minutes per side, until golden (it will not be cooked through) – do NOT let the drippings burn/blacken. Reduce heat if necessary. Remove chicken to a plate but do not discard drippings.
Reduce heat to medium. To the now empty skillet, melt 2 tablespoons butter with 2 tablespoons olive oil. Add onions, carrots, celery and mushrooms and cook until softened, about 10 minutes. Add a drizzle of olive oil if the vegetables become too dry. Add garlic and red pepper flakes and sauté for 30 seconds.
Add remaining flour from dredging chicken PLUS fresh flour as needed to equal 3 tablespoons total. Sprinkle flour over vegetables and cook for 2 minutes.
Reduce heat to low and slowly pour in the chicken broth while stirring and scraping up any brown bits on the bottom of the pan. Stir in dried parsley, dried oregano and dried basil. Return chicken to the pan along with any juices. Bring to a simmer, cover, and bake at 350 degrees F for 25-35 minutes, until the chicken reaches 170 degrees F.
Transfer the chicken from the skillet to a plate. Stir in heavy cream, fresh thyme and fresh parsley. Simmer until reduced to desired consistency, 5-10 minutes. Stir in lemon juice and salt and pepper to taste (I like 1/4 tsp each). Add chicken back to pot and serve over rice or mashed potatoes.