Preheat oven to 425°F (220°C). Toss squash and onion with half the olive oil, sage, salt and pepper. Roast for 20-25 minutes.
Slice chicken into cutlets. Mix Parmesan, panko, garlic and thyme. Dip chicken in beaten egg, then coat in Parmesan mixture.
Heat remaining oil in a skillet over medium heat. Cook chicken for 4-5 minutes per side until deeply golden and cooked through.
Slice chicken and serve with the caramelized squash and onion.
