Preheat oven to 425°F. Line a sheet pan with parchment paper.
Season chicken with salt and pepper.
Dredge each breast in flour, dip in beaten egg, then coat in the breadcrumb–Parmesan mixture.
Heat olive oil in a large skillet over medium heat. Pan-fry chicken 3 to 4 minutes per side until golden brown. Transfer to the sheet pan.
Taste onions and season with salt if needed. If too thick, stir in broth or wine to loosen.
Spoon about ¼ to ½ cup onion mixture over each cutlet.
Top with Gruyère and a sprinkle of Parmesan.
Bake 10 to 12 minutes, until cheese is melted and bubbly and chicken is cooked through (165°F internal temperature).
Garnish with more Parmesan and lots of fresh thyme sprigs and serve hot with salad or crusty bread.