Heat the olive oil in a large pot over medium heat.
Add the onion and cook until soft, about 5 minutes.
Stir in the garlic and cook for 1 minute.
Add the chopped broccoli and vegetable broth.
Bring to a boil, then reduce heat and simmer for 15 minutes until broccoli is tender.
Use an immersion blender to puree the soup until smooth.
Stir in the milk or cream and Stilton cheese until melted and creamy.
Season with salt and pepper to taste. Serve hot.