Bring a large pot of water to a boil over high heat, then salt it generously. Trim the root end of the leeks, leaving enough to hold the leaves together, and halve lengthwise. Rinse very well under running water to flush out any grit trapped between the leaves. Add the leeks to the boiling water and cook until tender, 7 to 10 minutes. Transfer the leeks to a paper towel-lined plate to drain. Pat dry and lay them on a serving platter.
Meanwhile, whisk the vinegar, mustard, water, and the 3/4 teaspoon salt in a small bowl. Gradually whisk in the olive oil, starting with a few drops and then adding the rest in a steady stream to make a smooth, slightly thick dressing. Drizzle the vinaigrette evenly over leeks, turn them to coat, and scatter the parsley and the tarragon, if using, over the top. Serve immediately or up to 2 hours later at room temperature.