Preheat the oven to 375°F (190°C).
In a large skillet, heat the butter over medium heat. Add the sliced onions and salt, and cook, stirring frequently, until the onions are deeply caramelized and golden brown, about 25 minutes.
Add the minced garlic and fresh thyme to the onions and cook for 1 more minute until fragrant. Remove from heat.
In a large mixing bowl, combine the caramelized onions, cooked chicken, beef broth, sour cream, 1/2 cup Gruyère cheese, 1/2 cup mozzarella cheese, Worcestershire sauce, olive oil, flour, black pepper, and dried parsley. Stir until well combined.
Transfer the mixture into a greased 9×13-inch casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup Gruyère cheese and 1/2 cup mozzarella cheese evenly over the top.
Bake uncovered in the preheated oven for 30 minutes until the casserole is bubbly and the cheese is melted and golden.
Remove the casserole from the oven and sprinkle the French fried onions evenly over the top.
Return the casserole to the oven and bake for an additional 5-10 minutes until the onions are crispy and golden brown.
Remove from oven and let cool for 5 minutes before serving.