Preheat the oven to 425°F (220°C). Position a rack in the center. Use an extra-large rimmed baking sheet to give the chicken and vegetables enough room to roast. If using two standard baking sheets, plan to distribute the ingredients between them rather than crowding everything onto one pan.

Place the cut baby potatoes and sliced carrots in a large mixing bowl. Add 1 tablespoon (15 mL) of the olive oil, ½ teaspoon of the kosher salt, and ¼ teaspoon of the black pepper. Toss until evenly coated. Spread the vegetables across the baking sheet in a single layer, concentrating them toward one side so there will be room for the chicken and broccoli later.

Place the baking sheet in the preheated oven and roast the potatoes and carrots for 15 minutes. The potatoes should begin developing golden edges, and the carrots should start to soften. Neither needs to be fully cooked at this stage.

While the vegetables roast, whisk together the Dijon mustard, honey, olive oil, apple cider vinegar, minced garlic, dried thyme, smoked paprika, and ½ teaspoon kosher salt in a small bowl. The glaze should be smooth, glossy, and thick enough to coat a spoon. Transfer 2 tablespoons of the glaze to a separate clean bowl and set it aside for serving. Keep this reserved portion away from the raw chicken.

Place the chicken pieces in a clean mixing bowl. Pour the remaining honey mustard glaze over the chicken and toss until every piece is thoroughly coated. The glaze should cling to the chicken without pooling excessively at the bottom of the bowl. Keep the chicken refrigerated if the vegetables are not yet ready for the next roasting stage.

Place the broccoli florets in a separate bowl: Add the remaining 1 tablespoon (15 mL) olive oil, ¼ teaspoon kosher salt, and ¼ teaspoon black pepper. Toss thoroughly, making sure the florets are lightly coated. Avoid adding too much oil, which can make the broccoli greasy rather than crisp-edged.

After the potatoes and carrots have roasted for 15 minutes, carefully remove the baking sheet from the oven. Turn the potatoes and carrots with a spatula. Add the glazed chicken and seasoned broccoli, distributing them across the remaining space. Arrange the chicken in a single layer with a little space between the pieces. Avoid piling the broccoli on top of the chicken, as this can trap steam and prevent proper browning. If the pan looks crowded, transfer some of the ingredients to a second baking sheet.

Return the baking sheet to the oven and roast for another 20–25 minutes. The chicken should develop golden, lightly caramelized edges, the potatoes should be fork-tender, and the broccoli should have browned tips with tender stems. Use an instant-read thermometer to check the thickest pieces of chicken. They must reach an internal temperature of 165°F (74°C) . If the chicken finishes before the potatoes, transfer it to a clean plate and keep it warm while the vegetables finish roasting. If the honey glaze begins darkening too quickly, move the baking sheet to a lower oven rack.

Remove the baking sheet from the oven and let the chicken rest for 5 minutes. Stir the reserved 2 tablespoons of honey mustard glaze, then drizzle it lightly over the cooked chicken and vegetables. Use clean tongs or a spatula to toss everything gently, keeping the roasted potatoes and broccoli intact.

Sprinkle the optional chopped parsley over the finished dish. Divide the chicken, potatoes, carrots, and broccoli evenly among five plates. Serve warm, spooning any cooked juices from the baking sheet over the portions.