Season chicken breasts with salt and pepper.
Set up a dredging station with three shallow bowls: flour in the first, beaten eggs in the second, and breadcrumbs mixed with Parmesan in the third.
Dredge each chicken breast in flour, then egg, then breadcrumb mixture, pressing gently so the coating sticks.
Heat olive oil in a large oven-safe skillet over medium heat. Add chicken and cook for 4 to 5 minutes per side, until golden brown and cooked through to an internal temperature of 165°F. Transfer to a plate.
In the same skillet, melt butter over medium heat. Add sliced onions and cook slowly, stirring occasionally, for 20 to 25 minutes until deeply caramelized.
Stir in garlic and cook for 1 minute. Add Worcestershire sauce and beef broth, scraping up any browned bits from the pan. Simmer for 2 to 3 minutes, until slightly reduced.
Return chicken to the skillet, nestling it into the onions. Spoon some of the onion mixture over the top of each piece.
Sprinkle Gruyère evenly over the chicken.
Transfer skillet to the oven and broil for 2 to 4 minutes, until the cheese is melted, bubbly, and lightly golden. Watch closely to prevent burning.
Finish with a sprinkle of fresh parsley. Serve immediately.