1/4 cup extra-virgin olive oil, divided, plus more for cooling rack
6 medium yellow squash or zucchini, or both (about 1 1/2 lbs. total), sliced into 1/2-in. rounds
3/4 teaspoon kosher salt, divided
1/4 teaspoon pepper
1-1/2 teaspoons curry powder
1 ½ teaspoons garlic powder
1 ½ teaspoons smoked paprika
1/8 teaspoon cayenne
1-1/2 pounds boned, skinned chicken thighs (5 to 8 pieces)
2 tablespoons honey
1 tablespoon Dijon mustard
1 tablespoon apple cider vinegar
Preheat broiler with oven rack 3 to 4 in. from heat. Set a metal cooling rack in a 12- by 17-in. rimmed baking sheet and coat lightly with oil.
Remove cooling rack and put squash on baking sheet. Sprinkle with 3 tablespoons oil, 1/4 teaspoon salt, and the pepper; toss to coat. Spread squash in a single layer and set cooling rack on top, nudging squash aside as needed so rack lies even.
In a large bowl, combine remaining 1/2 teaspoon salt with the curry powder, garlic powder, smoked paprika, and cayenne. Set aside 1-1/2 teaspoon spice mix.
Add remaining 1 tablespoon oil and the chicken thighs to large bowl and toss chicken to coat. Place spiced chicken smooth (skinned) side up on cooling rack over zucchini.
Broil chicken and squash, turning chicken once, until starting to char at edges, about 16 minutes total.
Meanwhile, make glaze: Whisk together reserved spice mix, honey, mustard, and vinegar in a small bowl.
Remove chicken and zucchini from oven. Generously brush chicken with glaze. Return to oven and broil, smooth side up, until nicely charred and an instant-read thermometer registers 165° when inserted in thickest part, about 4 minutes. Remove rack of chicken from baking sheet and set aside.
Return squash to oven and broil until most pieces are charred, 2 to 3 minutes more. Lift squash from juices and serve with chicken.