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1
In a small saucepan simmer reserved asparagus ends in water, covered, 15 minutes.
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2
Remove and discard asparagus ends with slotted spoon and bring water to a boil.
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3
Add reserved asparagus tips and cook, uncovered, over high heat until crisp-tender, about 3 minutes.
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4
Transfer asparagus tips with slotted spoon to a colander, reserving cooking liquid, and rinse under cold water to stop cooking.
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5
Drain tips well.
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6
In a 4-quart heavy saucepan cook onions in butter with salt and pepper to taste over moderate heat, stirring until pale golden.
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7
Add asparagus stalk pieces, broth, and reserved cooking liquid and simmer, covered, 15 minutes, or until asparagus pieces are tender.
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8
Make basil cream while soup simmers: In a small saucepan bring cream to a boil and stir in basil and salt.
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9
Cook mixture over high heat, stirring, until basil is wilted, about 5 to 10 seconds, and in a blender puree mixture.
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10
Return basil cream to small saucepan and keep warm.
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11
Puree soup in cleaned blender in small batches and return to 4-quart saucepan, thinning with water if desired.
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12
Season soup with salt and pepper and heat over moderately low heat, stirring occasionally, until heated through.
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13
Divide soup among 6 bowls and add asparagus tips, arranging them decoratively.
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14
Drizzle basil cream over each serving.
