Positions racks in the center and upper thirds of your oven, and preheat the oven to 425 degrees F.: You will notice the oven hum as it reaches temperature, and a properly preheated oven is essential for immediate searing and browning. The high heat encourages the fat under the skin to render quickly, producing crisp, blistered skin. If you skip preheating, the chicken will release juices slowly and may stew rather than roast, yielding limp skin and less concentrated flavor. A common mistake is setting the pan in too cool an oven, which prevents the Maillard reaction that creates color and depth.
Arrange the onion, apple slices, and Brussels sprouts in the center of a large, rimmed baking sheet. Top with 1 tablespoon oil, honey, mustard, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Toss to coat, then spread in an even layer.: As you toss, smell the sharp tang of mustard blending with honey, and see the glossy sheen on the vegetables indicating they will caramelize. Even layering ensures hot air circulates and each piece gets contact with the hot pan, which is where browning happens. Avoid piling ingredients on top of one another, because crowded vegetables steam and fail to brown properly. If you notice wet spots after tossing, spread the pieces so they dry slightly before roasting.
Nestle the chicken onto the sheet pan towards its center, skin side up, moving the apple and vegetables aside as needed so that the chicken sits directly on the sheet pan. Pat the chicken dry, and season with the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Drizzle the remaining 1 tablespoon of oil over the top. Lay a sprig of rosemary on top of each piece of chicken. (If you have more thighs than sprigs, cut the sprigs into pieces as needed and divvy them up.): The tactile act of patting the chicken dry will remove surface moisture, creating the conditions for crisp skin; you should hear a faint sizzle as the thighs contact the hot pan. Placing the chicken skin side up ensures the fat renders downward onto the vegetables for flavor. A common pitfall is leaving the skin damp which leads to steamed, rather than crisp, skin. If your rosemary sprigs are sparse, tearing them releases more aromatic oils but be mindful to remove woody stems before serving.
Bake the chicken and vegetables on the center rack for 15 minutes. Remove the pan from the oven, and use a spatula to stir the vegetables. Return the pan to the oven, and continue baking for 5 to 10 minutes, until the chicken is cooked through. It will register 165 degrees F on an instant-read thermometer inserted at the thickest part. If some pieces of the chicken finish sooner than others, remove them to a plate and continue cooking the remaining pieces until they are done. The apples will become very soft and break down, and the onions and Brussels sprouts will darken and become super crisp (this is the honey caramelizing).: During this time you will notice the kitchen filling with sweet, savory aromas, and the edges of the vegetables will brown and crisp. Stirring the vegetables after the initial roast redistributes caramelized bits and prevents burning in hot spots. Watch the chicken color, it should be deepening and juices should run clear when pierced. A frequent mistake is overbaking every piece to the point of dryness because some pieces finish earlier; remove done pieces as advised to preserve juiciness.
Remove the pan from the oven, then remove and discard the rosemary springs. Return any chicken pieces you removed earlier to the pan. Turn the oven to broil. Place the pan on the upper third of the oven and broil for 2 to 3 minutes, until the skin is golden and crispy. (This step will crisp the veggies further, too; if you prefer them not to continue crisping, remove them from the pan prior to broiling, or place the chicken on a different lined baking sheet.): The broil stage is dramatic, producing a sizzling chorus and visible bubbling on the skin as it crisps. Keep the oven door closed mostly, but watch through the window; broiling can go from golden to charred quickly. If you see blackened spots forming rapidly, pull the pan out immediately. A common error is walking away during broil and burning the skin, so stay attentive and use short bursts under the broiler if your oven runs hot.
Remove the pan from the oven, and season the chicken with an extra pinch of salt and chopped fresh rosemary. Serve the chicken hot with the apples and veggies.: After resting briefly, the finishing salt and fresh chopped rosemary will brighten the dish, and the contrast between warm, crispy skin and soft, jammy apples is especially pleasant. Taste one piece to adjust seasoning before serving everyone. A typical mistake is skipping the final sprinkle of salt which enhances and rounds the flavors; finishing touches matter for balance.