In a large pot, melt the butter over medium heat. Add the chopped onion and potato, and sauté until the onion becomes translucent.

Pour in the stock and add the chopped broccoli. Bring the mixture to a boil, then reduce the heat and let it simmer for about 20 minutes or until the vegetables are tender.

Using an immersion blender or a regular blender, puree the soup until it's smooth and creamy.

Return the soup to the pot and gently reheat it. Crumble in the Stilton cheese and stir until it's fully incorporated. Season with salt and pepper to taste.

Serve the soup hot, garnished with fresh chives and an extra sprinkle of Stilton, if desired.