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To make Parmesan Croutons: Preheat broiler.
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Adjust rack to 8 inches below broiler.
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Brush both sides of bread with oil, and place slices on a baking sheet.
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Broil for 3 to 10 minutes, turning as needed, or until golden brown and crisp.
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Remove from oven, and, when cool enough to handle, slice into cubes.
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Sprinkle with Parmesan cheese, and broil for 1 to 2 minutes more, or until golden.
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Remove from oven, and set aside.
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To make Asparagus Stock: Cut tips off asparagus, and set aside for garnish.
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Place asparagus spears in a soup pot with remaining ingredients.
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Bring to a boil over medium heat, reduce heat to low and cook, uncovered, for 20 minutes.
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Remove from heat, and drain into a bowl, reserving liquid.
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Discard solids.
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Measure 6 cups stock; if you have less, add enough water to measure 6 cups.
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To make Soup: While stock cooks, heat 2 tablespoons oil in a skillet over medium-low heat.
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Add half the onion, all potatoes and water.
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Cover, and cook for 20 minutes, stirring several times, until vegetables are soft.
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Cut tips off asparagus, and set aside with other tips for garnish.
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Cut asparagus spears into 1-inch-long pieces.
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Heat remaining 3 tablespoons oil in a large pot over medium heat.
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Add remaining onions and asparagus spear pieces, and cook, stirring frequently, for 3 to 5 minutes or until onions have softened.
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Add salt and strained Asparagus Stock, and bring to a slow boil.
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Cook over medium heat, uncovered, for 4 to 8 minutes, or until asparagus are tender, exact timing depending on size and freshness of asparagus.
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Using a slotted spoon, remove asparagus from pot, and put into a food processor.
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Process until smooth, adding stock as needed for processing.
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Transfer puree to a very fine strainer, and press through to remove tough fibers.
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Return puree to soup pot.
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Add potatoes and onions, and stir to mix.
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Add milk, salt and pepper, and cook over medium-low heat.
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Meanwhile, bring water to a boil.
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Add asparagus tips, and cook until just tender.
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Remove from heat, and strain.
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To serve, ladle soup into bowls, garnish each serving with asparagus tips and pass croutons, if using.
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Offer extra freshly ground black pepper.