Cut the broccoli into florets, peel the stems, and chop into approximately 1-inch cubes. Saute all vegetables in olive oil with salt and pepper, covered, until slightly softened and onion is translucent. Add four cups of vegetable stock and allow to come to a boil. Reduce heat and cover. Simmer until vegetables are very soft, about 20 to 30 minutes.

Melt butter in a separate pan. Add flour and whisk together to make a roux, cooking flour completely. Slowly whisk in remaining one cup of stock and half-and-half and allow to thicken slightly. Add shredded cheese and melt to create a cheese sauce.

Pour part of the vegetable and stock mixture into a blender. Place the lid on the blender, but remove the center and place a towel over the hole to allow steam to escape. Puree vegetables and stock until almost smooth. Do this in a few batches so you don’t overload your blender. This could also be done in a food processor or with an immersion blender.

Return pureed vegetables to the pot and combine with the cheese mixture. Stir in cream if desired and adjust seasoning. Serve with a sprinkle of shredded cheese on top.