Preheat the oven to 400°F (200°C).
Make the lemon garlic marinade: in a large bowl or large resealable bag combine 1/3 cup olive oil, 3 minced garlic cloves, 3 tablespoons lemon juice, 2 teaspoons lemon zest, 1 tablespoon Dijon mustard, 1 teaspoon salt, 1 teaspoon pepper, 1 teaspoon dried basil, 1 teaspoon dried oregano, 1 teaspoon onion powder, 1/2 teaspoon paprika, 1/2 teaspoon ground cumin, 1/2 teaspoon dried thyme, and 1/2 teaspoon sugar; mix to combine.
Reserve 2 tablespoons of the marinade (set aside in a small bowl); add the chicken thighs to the bag or bowl with the remaining marinade, seal or cover, and turn to coat. Let the chicken 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 spray with cooking spray.
Toss the quartered potatoes with 1 tablespoon of the reserved marinade (make sure this tablespoon never touched the raw chicken) and 1/4 teaspoon salt until evenly coated. Spread the potatoes in an even layer on one side of the prepared sheet pan and bake for 15 minutes at 400°F.
After 15 minutes, remove the pan from the oven. Push the potatoes to one side. Toss the broccoli florets with another 1 tablespoon of the reserved marinade and place them next to the potatoes in a single layer.
Arrange the marinated chicken thighs in a single layer on the empty side of the sheet pan.
Return the pan to the oven and bake uncovered for 17–22 minutes, or until chicken thighs reach 170–175°F (use an instant-read thermometer). If using chicken cutlets, cook until 165°F. Vegetables should be tender and lightly browned.
While the chicken bakes, 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 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 smooth. Chill until ready to serve.
Remove the pan from the oven when done, let the chicken rest a few minutes, then serve the chicken with potatoes and broccoli alongside the yogurt-Parmesan herb sauce.