Preheat oven to 425°F (220°C). Line a large rimmed sheet pan with parchment paper, lightly drizzle with olive oil, and place a wire rack on top if using.
Toss potato chunks with 1.5 tablespoons olive oil, half the minced garlic, 1/3 cup grated parmesan, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Spread in a single layer on the sheet pan and roast for 15-20 minutes.
Combine remaining 2/3 cup parmesan, panko, Italian seasoning, garlic powder, smoked paprika, onion powder, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a shallow dish.
In another bowl, whisk melted butter with remaining 1.5 tablespoons olive oil. Pat chicken completely dry, brush with butter-oil mixture, then press firmly into panko-parmesan coating to coat all sides.
Remove pan from oven, place breaded chicken on the rack or alongside potatoes, and spray tops with oil for extra crispiness.
Toss green beans with remaining minced garlic, a drizzle of olive oil, and a pinch of salt and pepper. Scatter around the potatoes and chicken in an even layer.
Return pan to oven and roast for 20-25 minutes until chicken reaches 165°F (74°C), potatoes are golden and tender, and green beans are tender-crisp. Optionally broil for 2-3 minutes for extra crunch.
Rest for 5 minutes before serving. Garnish with fresh parsley and serve with lemon wedges on the side.