In a large pot over medium heat, melt the butter. Add the diced onion, sliced leek, and minced garlic, and sauté until they become translucent and fragrant.
Add the chopped potato to the pot, stirring to combine with the onion mixture.
Pour in the vegetable stock and bring the mixture to a simmer. Let it simmer for about 10 minutes, or until the potatoes are tender.
Add the broccoli florets to the pot, and continue to simmer the mixture for another 10 minutes.
Once the broccoli is tender, remove the pot from the heat. Stir in the crumbled Stilton cheese and cream, then season with salt and pepper to taste.
Use an immersion blender to puree the soup until smooth, or transfer to a countertop blender in batches, if needed.
Return the soup to the heat to warm through, then ladle into bowls to serve.