Step 1

Arrange 2 racks to divide the oven into thirds and heat the oven to 425°F.

Step 2

Position 1 medium acorn squash on its side. Trim off both ends with a large, sturdy knife. Stand the squash upright and cut in half lengthwise. Using a spoon, scrape out the seeds and any stringy bits. Place the halves cut-side down and cut into 1/2-inch-thick half-moons. Transfer the squash to a rimmed baking sheet.

Step 3

Cut 1 medium red onion in 1/2-inch wedges and add it to the baking sheet. Drizzle with 2 tablespoons of the olive oil, and sprinkle with 1 teaspoon of the kosher salt and 1/4 teaspoon of the black pepper. Toss to coat, then arrange the squash and onion cut-side down in a single layer.

Step 4

Pat 4 bone-in, skin-on chicken thighs dry with paper towels. Use your fingers to gently loosen and pull the skin away from the meat without removing it. Tuck 2 fresh sage leaves under the skin of each thigh. Drizzle with the remaining 1 tablespoon olive oil and rub to coat. Season all over with the remaining 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper. Nestle the chicken skin-side up among the vegetables.

Step 5

Roast on the lower rack until the squash is tender and caramelized and the chicken is almost cooked through (registering about 155°F on an instant-read thermometer), 18 to 20 minutes.

Step 6

Meanwhile, smash and peel 1 garlic clove and place in a small saucepan. Add 1/2 cup balsamic vinegar and 1 tablespoon honey or maple syrup and bring to a boil over medium-high heat. Cook until syrupy and reduced by half, 5 to 8 minutes. Remove from heat and whisk in 1 tablespoon Dijon mustard.

Step 7

Remove the baking sheet from the oven and turn the oven to broil. Spoon some balsamic mixture onto each chicken thigh, using the back of the spoon to generously coat, then drizzle the remaining over the vegetables. Broil on the upper rack until the chicken is crispy in spots, golden-brown, and cooked through (registering 165°F), about 3 minutes. Serve the chicken and vegetables drizzled with the pan juices.