Heat olive oil in a large pot over medium heat.
Add diced onion, carrots, and celery. Cook for about 5 minutes until the vegetables soften.
Sprinkle flour over the vegetables and stir continuously for 1 minute to create a light roux.
Slowly pour in the chicken broth while stirring to prevent lumps.
Add thyme, salt, and black pepper for seasoning.
Stir in the shredded rotisserie chicken.
If using noodles, add egg noodles and cook until tender.
Reduce the heat and stir in milk or half-and-half.
Let the soup simmer for 10–15 minutes until thick and creamy.
Taste and adjust seasoning if needed. Serve hot.