Preheat the oven to 350°F (175°C).
In a large skillet, melt the butter over medium heat.
Add the sliced onions and salt to the skillet and cook, stirring frequently, until the onions are deeply caramelized and golden brown, about 25 minutes.
Add the minced garlic and dried thyme to the onions and cook for 1 more minute, stirring constantly. Remove from heat.
In a large mixing bowl, combine the rinsed rice and chicken broth.
Heat the olive oil in a separate skillet over medium-high heat.
Add the chicken pieces to the skillet and season with black pepper. Cook until the chicken is lightly browned on all sides but not fully cooked through, about 5 minutes. Remove from heat.
In a 9×13 inch casserole dish, spread half of the caramelized onion mixture evenly on the bottom.
Add the browned chicken pieces evenly over the onions.
Pour the rice and chicken broth mixture evenly over the chicken.
Top the casserole with the remaining caramelized onions.
Cover the casserole dish tightly with aluminum foil and bake in the preheated oven for 45 minutes.
Remove the foil and sprinkle the Gruyère, cheddar, and Parmesan cheeses evenly over the top.
Return the casserole to the oven uncovered and bake for an additional 10-15 minutes, until the cheese is melted and bubbly and the rice is tender.
Remove from the oven and let the casserole rest for 5 minutes before serving.