Preheat the oven to 400°F (200°C).
Make the lemon garlic marinade: in a large bowl or freezer-style sealable bag, combine 1/3 cup olive oil, minced garlic, 3 tablespoons lemon juice, lemon zest, Dijon mustard, 1 tsp salt, 1 tsp pepper, 1 tsp dried basil, 1 tsp dried oregano, 1 tsp onion powder, 1/2 tsp paprika, 1/2 tsp ground cumin, 1/2 tsp dried thyme, and 1/2 tsp sugar; mix until smooth.
Reserve 2 tablespoons of the marinade in a small bowl for the vegetables, then add the chicken thighs to the bag or bowl and toss to coat; seal and marinate at room temperature for 20 minutes or refrigerate up to 4 hours.
Line a large rimmed sheet pan (about 15×21 inches) with foil and lightly spray with nonstick cooking spray.
Toss the quartered baby potatoes with 1 tablespoon of the reserved marinade (never used on the raw chicken) and 1/4 teaspoon salt until evenly coated; spread the potatoes in an even layer on one side of the sheet pan and bake at 400°F for 15 minutes.
After 15 minutes, remove the pan. Push the potatoes to one side; add the broccoli to the pan, toss the broccoli with the remaining 1 tablespoon reserved marinade, and arrange it in a single layer next to the potatoes.
Arrange the marinated chicken thighs in a single layer on the pan alongside the vegetables.
Bake uncovered at 400°F for 17–22 minutes, until chicken thighs reach 170–175°F on an instant-read thermometer (if using chicken cutlets, cook to 165°F).
While the chicken bakes, make the yogurt Parmesan herb sauce: whisk together Greek yogurt or sour cream, 1/4 cup grated Parmesan, 1/4 cup chopped parsley, 1 tablespoon lemon juice, 1 tablespoon chopped chives (optional), 1 tablespoon minced dill (optional), 1/4 teaspoon salt, 1/4 teaspoon ground cumin, and 1/8 teaspoon pepper until combined; refrigerate until serving.
Once the chicken is cooked through, remove the pan from the oven and let the chicken rest a few minutes before serving with the roasted potatoes, broccoli, and yogurt Parmesan herb sauce.