Turn oven to 450 degrees Fahrenheit to preheat.
On high heat, heat a large skillet that’s ovenproof. Add in 1 T oil. Toss chicken with 1/4 tsp. salt and pepper. Place in pan. Cook until brown - about 3 minutes. Turn chick breast over and place pan in preheated oven. Bake for 9 minutes or until cooked through.
Remove pan from oven. Place chicken on plate and keep warm. Place the skillet over medium high heat. Add 1/2 C chicken broth and cider, bringing to a boil while using a spoon to loosen the browned bits on the bottom of the pan. Lower heat to medium-low and cook at a simmer for 4 minutes or until thick. Add mustard, 1 tablespoon butter, and parsley and whisk to mix.
In a large nonstick skillet, heat 1 T oil and 1 T butter over medium-high heat. Toss in Brussels sprouts and cook until lightly brown, about 2 minutes. Put remaining 1/8 tsp salt and 1/4 C broth in pan and cover. Cook 4 minutes.
Serve sprouts alongside the chicken and sauce.