Toss halved baby potatoes with 1 tablespoon olive oil, salt, and pepper on a sheet pan, then roast at 425°F.
Pound chicken breasts thin, dip in beaten egg, and coat with panko. Pan-fry in 3 tablespoons olive oil over medium-high heat, in 2 batches, about 5 minutes per side until golden and 165°F inside.
After potatoes have cooked 10 minutes, add green beans tossed with minced garlic and 1 tablespoon olive oil to the pan.
Return potatoes and green beans to the oven for 10 minutes until beans are tender and potatoes are crispy.
Plate chicken with vegetables.