Arrange the chicken on a cutting board and trim any gristle or tendons from the meat. Flatten the thicker part of the chicken with a meat mallet to create an even thickness. Season the chicken breasts with the salt and pepper. (Optional: Set aside and let rest for 40 minutes.) Dust lightly with 1 tablespoon of flour.

Heat 2 tablespoons of avocado oil in a large skillet over medium-low heat. Add the chicken to the skillet and sear for 3-4 minutes on each side until browned, about 8 minutes total. (Optional: Test doneness with an instant-read thermometer and remove chicken when it reaches 155°F internal temperature.) Transfer the chicken from the skillet to a platter and tent with foil. Pour off any extra oil in the skillet.

For the sauce, heat 2 tablespoons of butter in the same skillet over medium-low heat. Add the onions and cook until softened, about 5 minutes. Add the garlic and cook for 2 more minutes until fragrant.

Sprinkle 1 tablespoon of flour over the onions, stirring constantly until flour is incorporated into the butter. Continue stirring, gradually adding the chicken stock. Bring to a simmer, stirring occasionally until sauce thickens slightly, about 5 minutes.

Stir in the mustards, half-and-half, and any accumulated juices from the chicken into the simmering sauce. Simmer an additional 1-2 minutes until the sauce coats the back of a spoon. If needed, thin the sauce with chicken broth.

Slide the skillet off the heat and return the chicken to the skillet with the sauce. Place a lid on the skillet for 2-3 minutes to warm the chicken; garnish with parsley and serve.