Separate broccoli florets from stalks and set aside. Peel the stalks and dice. Peel the potato and dice. (You should have about 2 cups broccoli cubes and 1 cup potato cubes). Dice celery. Mince garlic to make 2 teaspoons.
Melt butter in a large pot and add cubed stalk, potato, celery and garlic. Saute a minute or two until the garlic is fragrant. Add the stock. Cover and bring to boil. Boil 5 minutes until tender.
Add florets and cover. Cook another 5 minutes until tender but still bright green. Blend soup to preferred texture.
Add half and half and cheese cubes and warm over low heat, stirring, until cheese begins to melt.
Serve with fresh herbs and/or croutons if you like