In a large pot or Dutch oven, melt the butter over medium heat. Add the onion, carrots, and celery. Cook for about 5 to 7 minutes, stirring occasionally, until the vegetables start to soften. Add the garlic and cook for another 30 seconds, just until fragrant.
Sprinkle the flour over the vegetables and stir well to coat everything evenly. Cook for about 1 minute. This helps remove the raw flour taste and gives the soup its creamy base.
Stir in the salt, pepper, and thyme. Slowly pour in the chicken broth while stirring to avoid lumps. Bring the soup to a gentle simmer.
Add the diced potato and cooked chicken. Let the soup simmer for about 15 to 20 minutes, or until the potatoes are fork-tender.
Stir in the milk and add the egg noodles. Simmer until the noodles are cooked according to the package directions. Taste and adjust seasoning if needed.
