Arrange a rack in the middle of the oven and heat the oven to 400°F. Meanwhile, sear the chicken.
Heat the oil in large cast iron or oven-safe skillet over medium-high heat until shimmering. Meanwhile, pat the chicken dry with paper towels and season with salt and pepper. Add the chicken skin-side down to the pan and sear until the fat is rendered and the skin is crisp and golden-brown, 6 to 8 minutes. Transfer the chicken skin-side up to a large plate. Remove all but one tablespoon of fat from the pan.
Add the shallots to the pan and cook over medium heat until softened about 3 minutes. Add the wine, scrape up any browned bits at the bottom of the pan with a wooden spoon, and cook until evaporated, about 3 minutes. Stir in the broth and whole-grain mustard and bring to a simmer. Return the chicken skin-side up, and add any juices accumulated on the plate to the pan.
Braise in the oven uncovered until the chicken is cooked through and reaches an internal temperature of 165°F, about 15 minutes. Transfer the chicken with tongs to a platter. Place the pan over medium-high heat, whisk the smooth Dijon mustard into the sauce, and simmer until reduced slightly, about 2 minutes.
Remove from the heat and whisk in the cream—taste and season the sauce with salt and pepper as needed. Serve the sauce with the chicken (we recommend serving it underneath the chicken to keep the skin crisp), and garnish with parsley if desired.
Storage: Leftovers can be stored in an airtight container in the refrigerator for up to 4 days.