Heat 2 tablespoons olive oil in a stock pot over medium heat. Cook and stir leeks in hot oil until completely softened, adding remaining olive oil in small amounts as you cook the leeks, about 10 minutes.

Pour vegetable broth and water into the pot; add potatoes. Bring mixture to a boil, reduce heat to medium-low, and cook at a simmer until the potatoes are completely tender, about 25 minutes.

Pour potato mixture into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until smooth. Stir cayenne pepper and salt into the soup.