How to Make It
Preheat oven to 425°F. Place a large baking sheet coated with cooking spray in the oven; heat 5 minutes.
Brush chicken with 1 tablespoon of the oil; sprinkle with 3/4 teaspoon each of the salt and pepper. Remove baking sheet from oven, and place chicken quarters, skin side down, on hot baking sheet; return to oven. Bake in preheated oven for 10 minutes.
Remove baking sheet from oven, and transfer chicken to a plate; add butternut squash, Brussels sprouts, onion, and garlic to baking sheet; drizzle with 1 1/2 tablespoons of the olive oil, and sprinkle with remaining salt and pepper. Toss to coat.
Place chicken on top of vegetables. Whisk together honey, Dijon mustard, and remaining 1/2 tablespoon oil; brush onto chicken using a pastry brush.
Bake at 425°F for 15 minutes. Baste vegetables and chicken with pan juices using a large spoon or bulb baster; sprinkle with rosemary. Return to oven, and bake until chicken is cooked through, about 20 minutes.