Preheat the oven to 375°F (190°C).
In a large skillet, melt the butter over medium heat. Add the sliced onions, sugar, salt, and pepper. Cook, stirring occasionally, until the onions are deeply caramelized and golden brown, about 25 minutes.
Add the minced garlic to the onions and cook for 1 minute until fragrant. Remove half of the caramelized onions and set aside for topping.
Add the chicken pieces to the skillet with the remaining onions. Sprinkle with dried thyme and rosemary. Cook, stirring occasionally, until the chicken is no longer pink on the outside, about 5 minutes.
Sprinkle the flour over the chicken and onions and stir to coat evenly. Cook for 2 minutes to remove the raw flour taste.
Slowly pour in the chicken broth and white wine, stirring constantly to avoid lumps. Bring to a simmer and cook until the sauce thickens slightly, about 5 minutes.
Remove the skillet from heat and stir in the sour cream, shredded Gruyère, mozzarella, and Parmesan cheeses until melted and smooth.
In a large mixing bowl, combine the cooked egg noodles with the cheesy chicken and onion mixture. Mix well.
Transfer the mixture to a greased 9×13-inch casserole dish. Spread the reserved caramelized onions evenly over the top.
Bake in the preheated oven for 20 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it rest for 5 minutes. Garnish with chopped fresh parsley before serving.