Preheat the oven to 200°C (400°F). Lightly grease a baking sheet with cooking spray or a thin coating of oil.
In a large bowl whisk the egg, lemon juice, 2 teaspoons minced garlic, chopped parsley, 1/2 teaspoon salt, and 1/2 teaspoon pepper until combined.
Place the chicken in the egg mixture to coat. Cover and refrigerate to marinate for 30–60 minutes if time allows (optional).
In a separate bowl combine the breadcrumbs and grated Parmesan.
Dredge each egg-coated chicken piece in the breadcrumb mixture, pressing gently so the coating adheres; set on the prepared baking sheet.
Arrange the quartered baby potatoes in a single layer around the chicken. In a small bowl mix half the melted butter with the remaining 2 teaspoons garlic and a pinch of salt, then pour this over the potatoes and toss to coat.
Bake in the preheated oven for 15 minutes.
Remove the tray, carefully flip each chicken piece, push potatoes to one side, and place the green beans on the other side of the sheet. Pour the remaining melted butter over the potatoes and vegetables.
Return to the oven and broil/grill on medium-high for about 10 minutes, or until the chicken is golden and crisp and the potatoes are cooked through. Check to avoid over-browning.
Sprinkle with extra chopped parsley if desired and serve immediately.