In a large resealable bag, combine the beef broth, white wine, 1 tablespoon of the mustard powder, Worcestershire sauce, chopped garlic, onions and some salt and pepper. Add the chicken quarters to the bag, seal well, and shake a bit to coat the chicken in the marinade. Marinate in the refrigerator for at least 1 hour, or up to overnight.
Preheat the oven to 400 degrees F
Heat the oil in a large skillet over medium-high heat. Remove the chicken from the marinade and pat dry with paper towels; reserve the marinade. Add the chicken, skin side down, to the skillet and cook until nicely brown, 5 minutes. Flip, and brown on the reverse side, another 4 minutes. Lower the heat to medium-low and add the butter and the smashed garlic to the pan. Cook for 1 minute, basting the chicken with the melted butter.
Transfer the skillet to the oven and roast until the chicken reaches an internal temperature of 170 degrees F, measured at the thickest part of the thigh, 10 to 12 minutes. Remove the chicken to a plate and let rest, loosely cover, for 5 minutes.
Heat the half-and-half and rosemary in a small saucepan over low heat. Pour off all but about 2 tablespoons of the oil from the skillet the chicken was cooked in. Place the skillet over medium heat and deglaze the pan with half of the reserved marinade, scraping up any browned bit from the bottom of the pan with a wooden spoon. Cook for 2 minutes, to reduce. Strain into the heated half-and-half and remove the rosemary sprig. In a small bowl, mix together the cornstarch and 2 tablespoons cold water. Whisk the cornstarch slurry into the half-and-half, then whisk in the remaining 1 1/2 tablespoons mustard powder, the whole-grain mustard and the lemon juice. Season the sauce with a salt and pepper to taste. Bring to a simmer and cook, stirring occasionally, until the sauce coats the back of a spoon. Pour some of the sauce over the chicken and serve the rest on the side.