First, preheat the oven to 375F degrees.
Season the chicken with the salt and pepper. Heat the duck fat in a cast iron skillet over medium high heat.
Add the chicken, skin side down and cook about 2-3 minutes until the skin is browned.
Remove chicken from the skillet and set aside. Turn the heat down to medium.
Add the shallots, garlic and thyme to the skillet and cook 1-2 minutes, until they just start to brown.
Add the wine to deglaze the pan and simmer for 2 minutes.
Add the broth and mustard and bring to a simmer. Cook for 2-4 minutes, remove from heat, and add the chicken thighs back in, skin side up.
Place skillet directly in the oven and cook for 30 minutes. The internal temperature should be 160F degrees when you remove it from the oven (it will heat the remaining 5 degrees when resting).
Let rest for 5 minutes, then serve drizzled with the sauce.