Heat the oven to 425°F with a rack in the middle. Line a large rimmed sheet pan with parchment paper, or lightly oil the pan if you prefer.

In a medium bowl, stir the miso paste, honey, and 2 tablespoons of the oil until smooth. If the mixture feels too thick to spoon easily, stir in 1 tablespoon warm water.

Put the chicken pieces in a large bowl. Spoon 3 tablespoons of the miso mixture over the chicken, then sprinkle with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper.

Toss the chicken in the bowl until the pieces are evenly coated. Set the bowl aside while you start the vegetables.

Put the carrots and red onion on the sheet pan. Drizzle them with the remaining 1 tablespoon oil.

Sprinkle the carrots and onion with 1/2 teaspoon of the salt and the remaining 1/4 teaspoon pepper. Toss them right on the sheet pan and spread them into a single layer.

Roast the carrots and onion for 10 minutes so the carrots get a head start.

While the first vegetables roast, put the broccoli florets and the scallion whites in a clean bowl. Add the remaining miso mixture and the remaining 1/2 teaspoon salt, then toss until the broccoli is lightly coated.

Pull the sheet pan from the oven. Use a spatula to move the carrots and onion toward the edges of the pan to make open space in the middle.

Put the chicken pieces in the center of the sheet pan. Leave a little room between the pieces so they brown instead of steam.

Add the broccoli and scallion whites to the open spaces around the chicken. Scrape any miso mixture left in the bowl over the broccoli.

Return the sheet pan to the oven. Roast for 12 minutes.