Preheat the broiler and set a rack about 6 inches from the heat.
Pat the chicken dry. Season all over with 1 teaspoon salt and the pepper.
Heat the olive oil in a large oven-safe skillet over medium-high heat.
Sear the chicken until golden on both sides, 6 to 8 minutes total. Transfer to a plate.
Reduce heat to medium. Melt butter in the skillet, then add the onions and remaining salt.
Cook, stirring often, until the onions are deeply golden and jammy, 12 to 15 minutes.
Stir in garlic and thyme. Cook until fragrant, about 30 seconds.
Sprinkle flour over the onions. Stir for 1 minute to coat and cook off raw taste.
Pour in the wine and scrape up browned bits. Simmer until reduced by about half.
Add broth, Worcestershire, and balsamic. Simmer until slightly thickened, 3 to 4 minutes.
Nestle chicken into the onions and sauce. Simmer gently until chicken reaches 165°F.
Scatter Gruyère over the chicken and onions.
Broil until the cheese melts and browns in spots, 1 to 3 minutes. Watch closely.
Rest 5 minutes. Sprinkle with parsley and serve hot.