Start by preheating your oven to 425°F. This high temperature ensures everything gets nice and crispy. Meanwhile, toss the red potatoes with olive oil, minced garlic, Parmesan, salt, and pepper in a mixing bowl. Spread them evenly over one-third of your sheet pan and pop it into the oven for 15 minutes. This gives the potatoes a head start since they take the longest to cook.
While the potatoes are roasting, prep the chicken. Mix breadcrumbs, Parmesan, salt, and pepper in one bowl and whisk a couple of eggs in another. Dip each chicken breast into the egg mixture, then coat it well with the breadcrumb mixture. Once the potatoes come out of the oven, arrange the chicken breasts on another third of the sheet pan.
Now it’s time for the green beans! Toss them in olive oil, garlic, salt, and pepper in the same bowl (no need to wash it). Scatter the seasoned beans across the final third of the sheet pan. Return everything to the oven.
Bake for another 20-25 minutes, or until the chicken reaches an internal temperature of 165°F. The potatoes should be golden brown, and the green beans will be tender yet crisp. For extra crispy potatoes or chicken, broil on high for 3-4 minutes at the end.