Season both sides of sliced chicken breasts with ½ teaspoon of salt and ½ teaspoon pepper.
Melt butter in a large OVEN SAFE skillet over medium high heat. Add onions and season with the remaining ½ teaspoon salt, ½ teaspoon pepper and thyme.
Cook, stirring often, (using a wooden spoon) until onions are browned and softened (about 10 to 11 minutes).
Add garlic to onions and cook (while continuing to stir) until fragrant, about 30 seconds.
Transfer onions with garlic to a large plate.
Heat olive oil in the same skillet over medium high heat. Add chicken and cook until golden brown, cooked through and no longer pink in the middle (about 4 minutes on each side depending upon thickness).
Transfer chicken to the plate with the onions.
Return onions with garlic to the skillet and sprinkle with flour. Stir well, while scraping any browned bits off of the bottom of the skillet for 1 minute.
Add beef broth and bring to a boil while stirring constantly.
Lower heat to maintain a simmer (medium low). Simmer until slightly thickened (about 1 ½ minutes), stirring occasionally.
Remove from heat and turn your oven broiler on high.
Carefully return chicken to the skillet and spoon gravy and some of the onions over the chicken.
Top each chicken breast with 1 slice of provolone cheese and sprinkle with shredded Gruyere.
Place under the broiler for 1 to 2 minutes, or until cheese is melted.