Preheat oven to 425°F. Line a large sheet pan with parchment paper or aluminum foil for easy cleanup.
In a small bowl, whisk together honey, 2 tablespoons olive oil, minced garlic, soy sauce, and paprika.
Place potatoes and carrots on the sheet pan. Drizzle with remaining 1 tablespoon olive oil, season with salt and pepper, and toss to coat. Spread into an even layer.
Arrange chicken thighs on top of the vegetables. Brush generously with half of the honey garlic mixture. Season with salt and pepper.
Roast for 25 minutes. Remove from oven and add broccoli florets to the pan. Brush chicken with remaining honey garlic sauce.
Return to oven and roast for an additional 10-12 minutes until chicken reaches 165°F and vegetables are tender. Let rest for 5 minutes before serving.