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.
Notes
Always cook with the skin side up in the oven to allow the skin to get crispier and remain crisp.
Don't cook the chicken past 165F degrees as it will be dry. I recommend taking it out at 160F degrees as the temperature will rise to 165F degrees while it rests.
Stir the garlic and shallots often as they will burn quickly. If they start to burn, just add the wine.
If you want a creamy sauce, substitute ¼ cup cream for ¼ cup of the chicken stock.
Substitute butter or bacon fat for the duck fat. You can also use olive oil in a pinch, but the flavor won't be as rich.
Drizzle with a touch of fresh lemon juice for added freshness and acid.