In a medium saucepan, cook potatoes in lightly salted water, covered, 10 to 15 minutes, until tender. In a large saucepan, combine the chicken stock (or dissolve bouillon cubes in water over medium heat) and broccoli. Cook, covered, about 10 minutes, until broccoli is very tender. Remove from heat. Using a large spoon, chop the broccoli into fairly small pieces against the side of the pot.
Drain the potatoes. Place about 1 cup potatoes on a saucer and mash well with a fork. (These will thicken the soup.) Add the mashed potatoes and remaining chopped cooked potatoes to the broccoli mixture, along with the milk and cream. Heat gently until hot but not boiling. Add the cheese and stir gently until melted and well blended. Season the soup with salt and pepper to taste; add the butter, if necessary.