Preheat your oven to 425 °F (220 °C). In a bowl, toss the chopped potatoes with garlic, olive oil, Parmesan, salt, and pepper until well coated. Spread them over about one third of the sheet pan. Roast for 10–15 minutes until they begin to become crisp.
While potatoes begin roasting, in the same bowl mix together olive oil, minced garlic, Parmesan, breadcrumbs, salt, and pepper. Press each chicken breast into that mixture to coat well. Remove the pan from the oven and slide chicken pieces next to the potatoes.
Toss the green beans with olive oil, garlic, (Parmesan if using), salt, and pepper. After chicken and potatoes have had a head start, place the beans on the sheet pan. Return the pan to the oven and roast for another ~25 minutes (or until chicken reaches 165 °F). For extra crispiness, broil 4–5 minutes at the end if desired. Let rest a few minutes before serving.