Preheat the oven to 350°F (175°C).
In a large skillet, heat olive oil over medium heat. Add the sliced yellow onion and cook, stirring frequently, until caramelized and golden brown, about 15 minutes.
Add the minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the rinsed rice, chicken broth, condensed French onion soup, dried thyme, salt, and black pepper. Stir until well mixed.
Add the caramelized onions and garlic to the rice mixture and stir to combine.
Lightly grease a 9×13 inch casserole dish with butter or cooking spray.
Spread the cut chicken pieces evenly in the bottom of the casserole dish.
Pour the rice and onion mixture over the chicken, spreading it evenly.
Cover the casserole dish tightly with aluminum foil.
Bake in the preheated oven for 40 minutes.
Remove the casserole from the oven and carefully remove the foil.
Sprinkle the shredded Swiss cheese evenly over the top of the casserole.
Return the casserole to the oven, uncovered, and bake for an additional 5 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.