Pat chicken dry; pound breasts to ¾-1 inch thickness if needed. Whisk together olive oil, lemon juice, oregano, garlic powder, onion powder, black pepper, and salt in a bowl or bag. Add chicken, coat well, and marinate in the refrigerator for at least 30 minutes (up to overnight for best flavor).

Preheat grill to medium-high (400-450°F / 200-230°C). Clean and lightly oil grates. Grill chicken for 5-7 minutes per side, until internal temperature reaches 165°F (74°C). Rest chicken for 5-10 minutes, then slice against the grain.

Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper. Wash and cut broccoli into bite-sized florets. In a large bowl, toss florets with olive oil, garlic powder, black pepper, and salt until evenly coated. Spread broccoli in a single layer on the baking sheet. Roast for 15-20 minutes, flipping halfway, until tender-crisp and caramelized.

In a small mixing bowl, combine mayonnaise, sour cream, minced garlic, chives, and lemon juice. Whisk vigorously until smooth. Add salt and black pepper; taste and adjust. Gradually whisk in milk if a thinner consistency is desired. For best flavor, cover and refrigerate for 15-30 minutes before serving.

Spoon cooked quinoa or brown rice into serving bowls. Arrange roasted broccoli and sliced grilled chicken. Generously drizzle with creamy garlic sauce. Garnish with fresh parsley or cilantro and extra lemon wedges if desired. Serve immediately.