Preheat oven to 200°C (400°F) and lightly grease a baking sheet with oil spray or a thin coating of oil.
Whisk together the egg, lemon juice, 2 teaspoons minced garlic, chopped parsley, 1/2 teaspoon salt, and 1/2 teaspoon pepper in a large bowl.
Dip each chicken piece into 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 the egg-coated chicken in the breadcrumb mixture, pressing gently so the crumbs adhere evenly.
Place the coated chicken on the prepared baking sheet. Arrange the quartered baby potatoes in a single layer around the chicken.
Mix the melted butter with 2 teaspoons minced garlic and a pinch of salt. Pour half the garlic butter over the potatoes and toss to coat.
Bake for 15 minutes in the preheated oven.
Remove the tray, carefully flip each chicken piece, push the potatoes to one side, and add the green beans to the other side. Pour the remaining garlic butter over the vegetables and return to the oven.
Broil/grill on medium-high for about 10 minutes, or until the chicken is golden and crisp and the potatoes are cooked through.
Sprinkle with additional chopped parsley if desired and serve immediately.
