Combine carrots, celery, onion, potatoes, salt, chicken base, and water in a 6 quart pot. Cook over medium high heat till tender, 10-15 minutes. Add broccoli during last five minutes of cooking time.
In a 2 quart saucepan, melt butter over medium heat. Whisk in flour till smooth. Whisk in about 2 1/2 cups of the milk. Whisk constantly over medium heat till sauce thickens and boils. Boil for at least one minute. Add to veggies.
Stir in cheese and remaining milk and heat over medium low heat till cheese melts.
Notes
The original recipe called for 3/4 cup butter, but I have been making it with less fat for years and no one ever misses it!
This makes a really chunky soup, so you can use less veggies if you prefer.