Preheat the Oven: Arrange a rack in the middle of the oven and preheat to 400°F.
Dry and Season the Chicken: Pat the chicken thighs thoroughly dry with paper towels, then season both sides with kosher salt and freshly ground black pepper.
Sear the Chicken: Heat the olive oil in a large cast iron or oven-safe skillet over medium-high heat until shimmering. Place the chicken skin-side down and cook for 6–8 minutes, until the fat has rendered and the skin is crisp and golden-brown.
Set Chicken Aside: Transfer the chicken to a large plate, keeping it skin-side up. Pour off the excess fat from the skillet, leaving about 1 tablespoon behind.
Cook the Shallots: Reduce the heat to medium, add the diced shallots, and cook for about 3 minutes, stirring as needed, until softened.
Deglaze with Wine: Pour in the dry white wine and use a wooden spoon to scrape up the browned bits from the bottom of the skillet. Cook for about 3 minutes, until the wine has evaporated.
Build the Mustard Sauce: Stir in the low-sodium chicken broth and whole-grain Dijon mustard. Bring the mixture to a simmer.
Return the Chicken: Place the chicken thighs back into the skillet, skin-side up, and pour any accumulated juices from the plate into the sauce.
Braise in the Oven: Transfer the uncovered skillet to the 400°F oven and braise for about 15 minutes, until the chicken is cooked through and reaches an internal temperature of 165°F.
Transfer the Chicken: Carefully remove the skillet from the oven and use tongs to transfer the chicken to a platter, keeping the skin-side up.
Reduce the Sauce: Place the skillet over medium-high heat. Whisk in the smooth Dijon mustard and simmer for about 2 minutes, allowing the sauce to reduce slightly.
Finish with Cream: Remove the skillet from the heat and whisk in the heavy cream until incorporated.
Season and Serve: Taste the sauce and adjust with salt and freshly ground black pepper as needed. Serve the sauce underneath the chicken to help keep the skin crisp, and garnish with coarsely chopped fresh parsley if desired.