Prepare a bowl of ice water. Trim and discard the stems of the asparagus (or save those ends for veggie broth). Cut the asparagus into 1-inch lengths. Set the tips aside. Place the sliced stems into a microwave-safe bowl with a cover. Microwave on high until the asparagus is bright green and just tender, about 2 minutes. Immediately plunge the asparagus into ice water to stop cooking. When cool, drain, and set aside with the tips.
In a large saucepan, heat the olive oil over medium heat. Add leeks and cook, stirring occasionally, until softened, about 3 minutes. Add the barley and stir to coat with oil. Cook, stirring occasionally, for 2 minutes. Stir in the wine and cook until it is absorbed. Stir in 2 cups of water and bring to a simmer. Cover and simmer very gently until the water is absorbed and the barley is tender (35-45 minutes). If the water is absorbed but the barley is not yet tender, stir in up to 1 cup water, cover, and cook until absorbed.
Stir in the reserved asparagus and salt. For the next 5 to 10 minutes, gradually stir in 1 cup of water. The barley should be creamy and tender. Remove the pan from the heat and stir in 3 tablespoons Parmesan, lemon zest, lemon juice, mint leaves and pepper. Add more lemon juice and pepper to taste. Divide the barley among four bowls.
Sprinkle 1 teaspoon Parmesan over each bowl and serve. Makes four servings.