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Cover eggs with cold water in a very small saucepan. Bring to a simmer and cook, covered, 5 minutes. Rinse eggs under cold running water to stop cooking, then peel and quarter.
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Cut asparagus on a very sharp diagonal into 3/4-inch-thick slices (about 2 inches long), leaving 2-inch tips. Halve tips lengthwise if thicker than rest of slices. Arrange asparagus on a steamer rack and sprinkle with kosher salt. Steam, covered, over boiling water until just tender, 2 to 3 minutes, then transfer to a bowl of ice water to stop cooking. Drain well and pat dry with paper towels.
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Whisk together vinegar and mustards. Add oil in a slow stream, whisking. Stir in shallot, 2 teaspoons tarragon, and salt and pepper to taste.
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Toss asparagus and celery leaves with half of vinaigrette and mound on 4 plates. Tuck quail eggs decoratively into salads. Drizzle salads and plates with remaining vinaigrette and sprinkle with remaining teaspoon tarragon.
