Preheat your oven to 375°F (190°C).
In a large skillet, heat the olive oil over medium heat. Add the sliced onions, salt, and pepper. Cook, stirring occasionally, until the onions are caramelized (about 15-20 minutes).
While the onions are cooking, season the chicken breasts with salt, pepper, garlic powder, and thyme. In another skillet, cook the chicken over medium heat until browned on both sides (about 6-8 minutes per side). Transfer to a cutting board and let it cool slightly before slicing into bite-sized pieces.
In the skillet with the caramelized onions, sprinkle the flour over the onions and stir. Then, add the beef broth and soy sauce. Bring to a simmer and cook for 5 minutes, stirring until the sauce thickens.
Remove the skillet from heat and mix in the sour cream and sliced chicken. Combine well and then pour the mixture into a greased casserole dish.
Top the casserole with the shredded Swiss and mozzarella cheeses, followed by the cubed French bread for added texture.
Bake in the preheated oven for 25-30 minutes, or until the cheese is bubbly and golden brown.
Allow the casserole to cool for a few minutes before serving.