Prepare the Honey Mustard Sauce: Whisk together the honey, Dijon mustard, whole-grain mustard, olive oil, minced garlic, paprika, dried thyme, salt, and black pepper in a mixing bowl until smooth and well combined.
Coat the Chicken: Pat the chicken thighs dry with paper towels. Add them to the honey mustard mixture and toss until each piece is thoroughly coated with the sauce.
Prepare the Vegetables: Wash and prepare the potatoes, carrots, and broccoli. Cut the vegetables into relatively similar-sized pieces so they roast evenly.
Arrange Everything on the Sheet Pan: Preheat the oven to 425°F. Lightly grease a large rimmed sheet pan. Arrange the chicken, potatoes, carrots, and broccoli in a single layer, leaving some space between the ingredients.
Roast Until Golden and Tender: Roast for about 30–35 minutes, or until the vegetables are tender and the chicken is golden and cooked through. The chicken should reach an internal temperature of 165°F in the thickest part.
Finish and Serve: Remove the sheet pan from the oven and let the chicken rest for about 5 minutes. Sprinkle with fresh parsley and serve warm with the roasted vegetables.