Combine the potatoes, celery, carrots, onion, and vegetable broth in a large pot. Bring to a boil, then cook over medium heat until potatoes are tender, about 10 to 15 minutes. Stir in the salt and pepper.

In a separate saucepan, melt the butter over medium-low heat. Add the flour and stir with a whisk, stirring constantly until thick, about 1 minute. Slowly stir in the milk to avoid lumps from forming. Continue stirring with a whisk over medium-low heat until thick, 4 to 5 minutes.

Stir the milk mixture into the potato mixture, and cook soup until heated through.

Serve immediately with your favorite toppings.