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Heat the oil in a large skillet over medium-high heat until shimmering. Meanwhile, pat the chicken dry with paper towels and season generously with salt and pepper. Add half of the chicken skin-side down to the pan and cook 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 6-quart or larger slow cooker. Repeat with browning the remaining chicken. Remove all but 2 tablespoons of fat from the pan.
Return the pan to medium heat. Add the shallots and cook 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 almost evaporated, about 6 minutes. Stir in the broth and whole-grain mustard and bring to a simmer. Pour over the chicken in the slow cooker.
Cover and cook on the LOW setting until the chicken is cooked through and reaches an internal temperature of 165°F, 4 to 6 hours (4 is preferable; at 6 hours the chicken on the bottom will start to get slightly mushy). Transfer the chicken with tongs to a platter and cover with aluminum foil to keep warm.
Turn the slow cooker to HIGH. Place the smooth Dijon mustard, cream, and cornstarch in a small bowl and whisk to combine. Whisk into the sauce in the slow cooker, cover, and cook until thickened, about 30 minutes. Taste and season the sauce with salt and pepper as needed. Serve the sauce with the chicken, and garnish with parsley if desired.