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1
Trim the ends off the asparagus and place in a pot of boiling salted water.
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2
Bring the water to a second boil, reduce the heat and boil slowly, uncovered, for 2 to 5 minutes, until the spears bend a little but are not limp.
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3
Drain and rinse them under cold running water until cool.
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4
Spread the cooked spears in a single layer on paper towels so they will finish cooling quickly.
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5
Place the eggs on a spoon and lower into a small saucepan of boiling water.
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6
Boil the eggs for 3 minutes (3 1/2 if they are chilled).
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7
Rinse under cold running water until cool enough to handle.
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8
Hit each egg with a knife at ''the equator'' and pull the halves apart.
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9
With a spoon, pull out the yolks into a medium-size bowl.
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10
Set aside the whites, in their shells, to finish cooking.
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11
Whisk the yolks until they are thick and sticky.
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12
Whisk in the mustard, salt and vinegar.
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13
Add the oil, whisking by hand or with an electric mixer, drop by drop at first and then in a steady stream.
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14
When all the oil is incorporated, stir in the shallots, parsley and basil or tarragon and taste for salt.
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15
Sieve or finely chop the egg whites and fold them in, if desired.
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16
If sauce is too thick for dipping, thin with a little vinegar.
