Heat olive oil in a large deep skillet or Dutch oven over medium heat.
Add diced onion, minced garlic, sliced carrots, and sliced celery to the skillet. Sauté for 5-7 minutes until vegetables are softened.
Sprinkle dried thyme, dried rosemary, black pepper, and salt over the vegetables. Stir to combine.
Add the butter to the skillet and let it melt completely.
Sprinkle the flour over the vegetables and butter mixture. Stir continuously for 2 minutes to cook the flour and form a roux.
Slowly pour in the chicken broth while stirring constantly to avoid lumps. Then add the whole milk and continue stirring until the mixture is smooth.
Bring the liquid to a gentle boil, then add the uncooked egg noodles and shredded chicken to the pot.
Reduce heat to medium-low and simmer uncovered for 12-15 minutes, stirring occasionally, until the noodles are tender and most of the liquid is absorbed.
Remove the skillet from heat and stir in the shredded cheddar cheese, mozzarella cheese, and Parmesan cheese until melted and well combined.
Let the casserole sit for 5 minutes before serving to thicken slightly.