Add the flour, salt, black pepper, thyme, and oregano to the softened vegetables. Stir well and cook for 2 minutes to form a roux and release the flavors.
Pour in the chicken broth and add the diced potato. Stir briefly to combine. Increase the heat to medium-high and bring to a boil without stirring. Boil for 3 minutes.
Reduce the heat to medium-low, partially cover the pot, and simmer for 25 minutes until potatoes are tender. Taste and adjust seasoning if needed.
Stir in the cooked chicken, half-and-half or whole milk, and uncooked noodles. Continue cooking for 10 minutes, stirring occasionally, until noodles are al dente and the soup has thickened. Adjust seasoning as preferred.
Ladle into bowls, garnish with fresh thyme leaves if desired, and serve warm.
Cool completely before transferring leftovers to an airtight container. Refrigerate for up to 1 week. Reheat gently on the stove over medium heat, adding more chicken broth if the soup is too thick.