Season the chicken breasts with salt and pepper on both sides.
In a large oven-safe skillet, heat the butter and olive oil over medium heat. Add the chicken breasts and cook for about 4-5 minutes on each side, until browned. Set aside.
Peel the onions, trim the ends, and cut them in half. Slice them lengthwise. In the same skillet (don't wash it, the brown bits add flavor), add the sliced onions. Add salt, pepper, and sugar. Cook, stirring occasionally, for about 10 minutes. Be patient and let them cook slowly to develop their sweetness.
Add the white wine to the onions and let it cook for a couple of minutes to let the alcohol and acidity evaporate. Cover the skillet and cook for 15 minutes until the onions are golden brown and caramelized.
Preheat your oven to 375°F (190°C)
Sprinkle the corn flour over the onions, then add the chicken broth and thyme. Let it simmer, stirring constantly, for a few minutes until the sauce has thickened slightly.
Add the chicken breasts back to the skillet, nestling them into the caramelized onions. Sprinkle a generous amount of shredded cheese over the top of each chicken breast. Transfer the skillet to the preheated oven and broil for about 10 minutes, or until the chicken is cooked through and the cheese is melted and bubbly
Remove from the oven and let the dish rest for a few minutes before serving. Serve with a side of your choice, such as a green salad or roasted vegetables.