Pat chicken dry. Season both sides with 1 teaspoon kosher salt, ½ teaspoon black pepper, garlic powder, smoked paprika, and onion powder.
Heat a 12-inch cast iron skillet over medium-high. Add 1 tablespoon avocado oil; when shimmering, cook chicken skin-side down 7–8 minutes until deeply golden. Flip and cook 2 minutes more. Transfer to a plate.
Place potatoes cut-side down in rendered fat. Drizzle with 1 tablespoon oil; season with ¼ teaspoon salt and ⅛ teaspoon pepper. Cook undisturbed 4–5 minutes until golden.
Scatter broccoli around potatoes: add remaining 1 tablespoon oil, remaining ¼ teaspoon salt, and remaining ⅛ teaspoon pepper. Toss and nestle chicken skin-side up.
Roast 18–20 minutes until chicken reaches 175–185°F, potatoes are fork-tender, and broccoli is lightly caramelized.
Whisk honey mustard sauce ingredients until smooth.
Brush half the sauce on hot chicken. Drizzle the rest over potatoes and broccoli or serve for dipping. Garnish with thyme, parsley, and lemon. Serve.