Preheat oven to 425°F with racks in upper and lower third positions. Arrange squash halves, cut-side down, on a large baking sheet. Roast on the upper rack until starting to soften, about 25 minutes.

Meanwhile, whisk 3 tablespoons oil, 2½ tablespoons maple syrup, 2 tablespoons miso, 1 tablespoon sesame oil, the grated garlic and grated ginger, ½ teaspoon soy sauce and ¼ teaspoon salt in a large bowl. Transfer 2 tablespoons of the mixture to another large bowl and set aside. Add trimmed chicken thighs and halved Brussels sprouts to the remaining glaze and toss to coat. Arrange the chicken, skinned-side up, on another large rimmed baking sheet and spread the Brussels sprouts, cut-side down, around the chicken.

Without removing the squash from the oven, place the sheet with chicken and Brussels sprouts on the lower third rack. Roast until the squash is tender, the chicken is browned, and an instant-read thermometer inserted in the thickest part of the chicken registers 165°F, about 15 minutes.

Remove the squash from the oven and let cool slightly. Move the chicken and Brussels sprouts to the upper rack and increase the oven temperature to broil. Broil until browned, 4 to 5 minutes. Sprinkle with the remaining ⅛ teaspoon salt.

Using a fork, scrape the squash flesh into strands (discard skins) and transfer to the bowl with the reserved glaze; toss to coat. Divide the squash, chicken and Brussels sprouts among 4 plates. Top with sliced scallion and 1 teaspoon sesame seeds.