Cook the Vegetables: In a large pot or Dutch oven (at least 4-quart capacity), melt the butter over medium heat. Add the onion, carrots, celery, and garlic. Stir occasionally, cooking for 5 minutes or until the vegetables soften. Stir in flour, salt, pepper, thyme, and oregano, and cook for an additional 2 minutes.
Add Broth and Potatoes: Pour in the chicken broth and add the diced potatoes. Stir briefly, then increase the heat to medium-high. Bring the soup to a boil, uncovered, and cook for 3 minutes. Lower the heat to medium-low, partially cover, and simmer for about 25 minutes, or until the potatoes are tender. Taste the soup and adjust the seasonings as needed.
Add Chicken, Milk, and Noodles: Stir in the shredded chicken, half-and-half (or milk), and uncooked noodles. Cook for about 10 minutes, until the noodles are tender and the soup has thickened. Taste and adjust seasoning again if necessary.
Serve: Ladle the soup into bowls and garnish with fresh thyme leaves, if desired. Serve warm.