In a large pot, combine vegetable stock, water, and broccoli. Bring to a boil, then reduce heat and simmer for 10 minutes, or until broccoli is tender.

Remove from heat and transfer the soup to a blender or food processor. Blend until smooth.

Return the soup to the pot and stir in the lemon juice.

While the soup is cooking, prepare the pesto. Place the nuts into a food processor and grind into crumbs. Add the olive oil, garlic, salt, and pepper. Process until blended into a course, pesto-like consistency.

Serve the soup in bowls with a tablespoon of pesto stirred in.