Preheat the oven to 350°F (175°C).
In a large skillet, melt the butter over medium heat.
Add the sliced onions, salt, and pepper to the skillet and cook, stirring occasionally, until the onions are deeply caramelized and golden brown, about 25 minutes.
Sprinkle the flour over the caramelized onions and stir well to combine; cook for 1-2 minutes to remove the raw flour taste.
Slowly add the chicken broth to the skillet, stirring constantly until the mixture thickens slightly.
In a large mixing bowl, combine the sour cream, mayonnaise, shredded cheddar cheese, shredded Swiss cheese, grated Parmesan cheese, dried thyme, and the caramelized onion mixture. Stir until fully combined.
Add the shredded cooked chicken to the mixture and fold gently to combine.
Transfer the mixture to a greased 9×13-inch casserole dish and spread evenly.
Sprinkle 1 cup of French fried onions evenly over the top of the casserole.
Bake in the preheated oven for 25 minutes, until the casserole is hot and bubbly and the onions on top are golden brown.
Remove from the oven and sprinkle the remaining 1 cup of French fried onions on top for extra crunch.
Let the casserole rest for 5 minutes before serving.