Preheat the oven to 400°F (200°C). Line a large sheet pan with parchment paper or aluminum foil for easy cleanup.
Wash and dry the potatoes thoroughly. Cut them into evenly sized bite-sized pieces. Place them in a bowl, drizzle with olive oil, season with salt, pepper, and a pinch of paprika, and toss to coat.
Spread the potatoes on one side of the sheet pan in a single layer. Roast them for 15 minutes to give them a head start.
While the potatoes are roasting, prepare the chicken coating. In a bowl, combine grated parmesan cheese, breadcrumbs, Italian seasoning, paprika, salt, pepper, and minced garlic.
Pat the chicken breasts dry with paper towels. Lightly brush or drizzle them with olive oil, then press them into the parmesan mixture, coating both sides evenly.
Trim the green beans and place them in a bowl. Toss with olive oil, salt, and pepper until evenly coated.
Remove the sheet pan from the oven after the potatoes have partially cooked. Push the potatoes slightly to make space for the chicken and green beans.
Place the parmesan-coated chicken breasts in the center of the pan and arrange the green beans on the remaining space.
Return the sheet pan to the oven and bake for an additional 20–25 minutes, or until the chicken reaches an internal temperature of 165°F (74°C), the crust is golden, and the vegetables are tender.
For extra crispness, switch the oven to broil for 2–3 minutes at the end, keeping a close eye to prevent burning.
Remove from the oven and allow the chicken to rest for 5 minutes before serving.