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Preheat oven to 400 degrees F.
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Make your pistachio cream. Combine 1 cup vegetable broth and next 3 ingredients, through lemon juice in a blender. Puree until smooth.
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Roast your asparagus. Line a baking sheet with parchment paper. Cut off the asparagus tips and spread evenly on the lined baking sheet. Drizzle 1 Tablespoon olive oil onto asparagus and toss with tongs until evenly coated. Roast for 10 minutes. Place asparagus tips to the side for serving.
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Meanwhile, start your soup. Heat remaining 2 Tablespoons olive oil in a large soup pot over medium high heat. Add the onion and leek and saute until they begin to soften, about 3 minutes. Add potato and season with salt and pepper. Saute until onions are browned and potato is soft, about 10 minutes. Add garlic and stir to combine.
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Add 1 cup of vegetable broth, and scrape the bottom of the pot to loosen browned bits. Add remaining 5 cups of broth, and asparagus stalks, and bring to a boil. Simmer for 5 minutes. Using an immersion blender, blend until smooth. Stir in 1 cup of the pistachio cream.
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Serve with remaining pistachio cream, asparagus tips, roasted pistachios, mint leaves, and lemon slices.