Soften the Aromatics: Melt the butter in a large saucepan over medium heat. Add the onion, leek and garlic and cook gently until soft and translucent, without allowing them to brown.
Cook the Potato: Add the diced potato and vegetable stock. Bring to a gentle simmer and cook for about 10 minutes to give the potato a head start.
Cook the Broccoli: Add the broccoli and continue to simmer for another 8–10 minutes, until the broccoli and potato are completely tender.
Blend the Soup: Remove the saucepan from the heat and blend with an immersion blender until completely smooth. Alternatively, carefully blend in batches using a countertop blender.
Add the Stilton: Stir the crumbled Stilton and cream into the hot blended soup until the cheese has completely melted.
Season and Serve: Taste the soup before seasoning, then add freshly ground black pepper and salt only if needed. Return to low heat briefly to warm through if necessary, but don't allow the soup to boil. Ladle into bowls and serve.