Melt the butter in a large soup pot over medium heat.
Add the chopped onion, celery, and carrots.
Cook, stirring occasionally, until the vegetables are soft.
Sprinkle the flour over the cooked vegetables.
Stir well to combine until the flour is fully mixed in.
This usually takes about 1-2 minutes.
Gradually pour in the chicken broth while stirring continuously.
Bring the mixture to a boil.
Let it simmer until it starts thickening.
Add the cooked, shredded chicken to the pot.
Stir in the milk, salt, and black pepper.
Keep simmering until everything is heated through.
Taste and adjust seasonings if needed.
Serve hot.
For a richer flavor, you can use cream instead of milk!