Melt butter over medium heat in a large saucepan or dutch oven. Add the celery, onion, thyme, salt and pepper. Saute until onion is tender--about 5 minutes.
Add the stock cubes, broccoli and 3 cups of water. Bring to boil, cover and reduce heat to low. Simmer 15 minutes.
Mix the cornstarch and the 1/2 cup water.This is a LOT of cornstarch but you have to end up with the consistency of a pudding so you must use a lot. Stir in the cornstarch and when bubbles begin to rise to the surface, slowly add the heavy cream, blending well. Stirring constantly, add the shredded cheese and stir until melted.
If the consistency is too thick for you taste, you can thin to the desired consistency using the *milk. Use just enough milk to get it to the consistency you prefer. I use about 1/4 cup of milk, but mileage may vary.
Serve immediately (as thickened products that use a cornstarch base will break down quickly).
Serves about 6.