Preheat the oven to 400°F.

To make the pistachio cream, put the water, pistachios, mint, lemon juice, and 1/4 teaspoon of salt in a blender or food processor and process until smooth and creamy. Taste; you may want to add a bit more salt.

Cut off the asparagus tips and put the stalks and tips on a rimmed baking sheet in a single layer. Drizzle the asparagus with 1 tablespoon of the olive oil and sprinkle with 1/4 teaspoon of salt. Toss evenly to coat. Bake for 8 to 10 minutes, until tender. Remove the asparagus tips from the baking sheet and reserve for garnishing the soup.

Heat the remaining 2 tablespoons of olive oil in an 8-quart pot over medium heat. Add the onion, leeks, and a pinch of salt. Sauté for 4 minutes, then add the potato and a generous pinch of salt. Stir occasionally, allowing the potatoes to soften and the onion to turn golden. Add the garlic and sauté for 30 seconds. Pour in 1 cup of the broth and deglaze the pot, stirring to loosen any bits stuck to the pot. Add the remaining 5 cups of broth and bring to a boil. Decrease the heat to low and simmer for 5 minutes.

In batches, puree the soup in a blender, adding the broth first, then the vegetables and asparagus. Blend until velvety smooth, about 2 minutes. If the soup is too thick, add more broth ½ cup at a time. Return the soup to the pot and gently reheat. Taste; you may want to add a pinch of salt or a squeeze of lemon.

Stir 1 cup of the pistachio cream into the soup. Ladle the soup into bowls and garnish with the reserved asparagus tips and parmesan cheese.