Heat the butter and olive oil in a large heavy-bottomed pot or Dutch oven over medium heat. Add the chopped onion and cook until softened and translucent, about 5 minutes.

Add the minced garlic, diced broccoli stems, and diced potato to the pot. Sauté for another 3 to 4 minutes, stirring frequently, until the vegetables start to release their aromas.

Pour in the vegetable broth and bring the mixture to a rolling boil over medium-high heat. Once boiling, reduce the heat to low, cover the pot, and simmer for 10 minutes.

Stir in the broccoli florets and continue to simmer, covered, for another 5 to 8 minutes. Test the vegetables with a fork; they should be very tender and easily pierced.

Remove the pot from the heat and carefully blend the soup using an immersion blender until completely smooth and velvety. Alternatively, transfer the soup in batches to a standing blender, pureeing until smooth before returning to the pot.

Return the blended soup to low heat. Stir in the crumbled Stilton cheese and heavy cream, cooking gently just until the cheese is fully melted and incorporated. Do not let the soup boil at this stage, or the cheese may split.

Taste the soup and adjust the seasoning with salt and black pepper, keeping in mind that the Stilton is naturally salty. Ladle the hot soup into warm bowls, garnish with a few extra crumbles of cheese, and serve immediately with crusty bread.