Heat oven to 425°F; line a large sheet pan with parchment or foil.
Whisk honey, 2 tablespoons oil, garlic, soy sauce, and paprika in a small bowl.
Toss potatoes and carrots on sheet with remaining 1 tablespoon oil, salt, and pepper. Spread evenly.
Arrange thighs over vegetables; brush generously with half the honey mixture. Season with salt and pepper.
Roast 25 minutes. Add broccoli; brush chicken with remaining sauce.
Roast 10-12 minutes more until chicken reaches 165°F and vegetables are tender. Rest 5 minutes before serving.