Melt the butter in a large pot or Dutch oven over medium heat.
Add the onion, celery, and carrots to the pot. Cook until the vegetables soften, about 5 to 7 minutes.
Stir in the minced garlic and cook for 1 minute until fragrant.
Whisk the flour into the vegetable mixture. Cook for 1 minute, stirring constantly, to create a roux.
Gradually whisk in the chicken broth until the mixture is smooth and there are no lumps.
Bring the mixture to a simmer. Add the milk, heavy cream, thyme, salt, and pepper. Stir well.
Add the shredded chicken and the egg noodles to the soup. Continue to simmer until the noodles are tender, about 8 to 10 minutes. Stir occasionally to prevent sticking.
Taste the soup and adjust salt and pepper as needed before serving hot.