Preheat the oven to 400°F (200°C).
Make the lemon-garlic marinade: whisk together 1/3 cup olive oil, 3 minced garlic cloves, 3 tbsp lemon juice (reserve 2 tbsp aside), 2 tsp lemon zest, 1 tbsp Dijon mustard, 1 tsp each salt, pepper, dried basil, dried oregano, and onion powder, plus 1/2 tsp each paprika, ground cumin, dried thyme, and sugar.
Place the marinade in a large resealable bag and remove 2 tablespoons (label as reserved for veggies); add the chicken thighs to the bag, seal, and turn to coat. Let chicken marinate at room temperature 20 minutes or refrigerate up to 4 hours.
Line a large sheet pan (about 15×21 inches) with foil and spray with cooking spray.
Toss the quartered potatoes with 1 tablespoon of the reserved marinade and 1/4 teaspoon salt, spread into an even layer on one side of the sheet pan, and bake for 15 minutes at 400°F.
After 15 minutes, remove the pan. Push the partially cooked potatoes to one side, add the broccoli and toss with the remaining 1 tablespoon reserved marinade, then arrange the broccoli next to the potatoes in a single layer.
Arrange the marinated chicken thighs in a single layer on the pan alongside the vegetables.
Bake uncovered at 400°F for 17–22 minutes, or until chicken thighs register 170–175°F on an instant-read thermometer (chicken cutlets 165°F).
Meanwhile, make the yogurt-Parmesan herb sauce: whisk together 1 cup Greek yogurt or sour cream, 1/4 cup finely grated Parmesan, 1/4 cup chopped parsley, 1 tbsp lemon juice, 1 tbsp chopped chives (optional), 1 tbsp minced dill (optional), 1/4 tsp salt, 1/4 tsp ground cumin, and 1/8 tsp pepper. Chill until ready to serve.
When the chicken reaches the target temperature, remove the pan from the oven and let the chicken rest a few minutes before serving with the roasted potatoes, broccoli, and the yogurt-Parmesan herb sauce.