In a large skillet over medium heat, heat 2 tablespoons 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, 1 minute more. Turn off heat and remove onion mixture. Wipe skillet clean.
In a large bowl, season chicken with salt, pepper and oregano, then toss with flour. Heat remaining oil in 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 caramelized onions to skillet. Bring mixture to a boil, then reduce heat and simmer until chicken is cooked through and beef broth reduces slightly, about 10 more minutes.
Add Gruyère and cover skillet with a lid. Cook until cheese is melty, about 2 minutes. Remove from heat and garnish with parsley before serving.
