In a large skillet over medium heat, heat 2 tbsp olive oil. Add onions and season with salt, pepper, and thyme. Reduce heat to medium-low and cook, stirring occasionally until onions are caramelized and jammy, about 25 minutes. Stir in garlic and cook until fragrant, about 1 minute. Turn off heat and remove onion mixture to a separate bowl. Wipe skillet clean.
In a large bowl, season chicken with salt, pepper and oregano, then toss with flour. Heat remaining oil in the same skillet over medium high heat. Add chicken and cook until golden on all sides and mostly cooked through, about 8 minutes.
Add beef broth and return cooked onions to skillet. Bring mixture to a boil, then then reduce heat and simmer until chicken is cooked through and beef broth reduces slightly, about 10 more minutes. Stir in cornstarch/water mixture. Simmer for about 2 minutes.
Add Gruyère and cover skillet with a lid. Cook until cheese is melty, about 2 minutes. Remove from heat and garnish with parsley. Serve warm. (Alternatively, sprinkle gruyere over the top and broil in preheated oven uncovered for about 1 minute.).