Remove and discard tough lower stalk of each broccoli. Peel remaining stalk. Finely chop stalk and florets (you should have about 4 cups). Set aside.
Heat oil in a medium heavy pot over medium heat. Add onion and sweat until soft and translucent, about 10 minutes. Add stock and potato and bring to a boil. Reduce heat and simmer until potato is almost tender, about 12 minutes. Stir in chopped florets. Simmer until potato and broccoli are very tender, about 10 minutes. Remove soup from heat and purée in a blender until smooth. Season to taste with salt and pepper. Serve hot with crusty bread.