Whisk together olive oil, vinegar, tarragon, mustard, pepper, sugar, shallot, and 3/4 teaspoon of the salt.
Bring water and remaining 3 tablespoons salt to a boil in a large saucepan over medium-high. Remove and discard tough ends of asparagus spears. Place asparagus in boiling water; cook until tender-crisp, 2 to 3 minutes. Drain and rinse with cool water; pat dry.
Arrange asparagus on a serving platter.
Drizzle with vinaigrette, and toss lightly to coat.
Top with grated eggs. Serve warm or at room temperature.