Sauté Onion: Heat two tablespoons of butter in a large soup pot over medium-high heat, add onion and sauté for 5 minutes or until translucent.
Add Garlic: Stir through the minced garlic and sauté for 1 minute.
Cook Leeks: Add leeks and the remaining 1 tablespoon of butter and cook for 5 minutes until tender and translucent, maintaining medium-high heat.
Incorporate Potatoes: Add the cubed potatoes and cook for further 5 minutes, stirring well.
Add Liquid: Pour in the vegetable broth, it should be enough to just cover the vegetables. Season with salt and pepper.
Bring to Boil: Increase the heat, put the lid on, and bring to a boil. Once boiling, reduce the heat to low and simmer for 30 minutes, with the lid on, until potatoes are tender.
Blend Soup: Remove from the heat and blend the soup in batches in a food processor or blender until smooth and thick. Or use an immersion blender directly in the soup pot.
Finalize Soup: Return the blended soup to the pot. Over medium heat, stir in the cayenne pepper and season with salt and pepper once again. Bring to a boil, then remove from the heat.
Serve: Garnish with fresh parsley and enjoy!