Bring a large, heavy pot of water to a boil. Add the broccoli and boil for five minutes. Drain the broccoli well and set aside.
Add the olive oil and garlic to the pot over medium heat. After a minute or two, when the garlic starts to soften and turn golden, add the broccoli, season with salt and pepper, and stir well.
Cover the pot, turn the heat down as low as it will go, and cook for about an hour, stirring occasionally, until the broccoli is soft enough that it yields when you press it with the back of a wooden spoon (it may brown a little during this process -- this is a good thing).
Add the chicken stock and bring to a boil over medium-high heat. Simmer the soup for 5 minutes.
Carefully puree half the soup in a blender or food processor, using a kitchen towel to hold the lid on tight. Stir the puree back into the pot. Stir in the Parmesan and lemon juice to taste. Taste and adjust the seasoning. Serve hot with plenty of crusty bread.