Preheat oven to 350°. Toast walnuts on a rimmed baking sheet, tossing once, until golden brown, 6–8 minutes. Let cool, then coarsely chop. Transfer to a small bowl and toss with oil; season with salt.

Trim root ends of leeks and 1″ from the dark green tops. Cut leeks into ½”-thick rounds (the dark green parts won’t be rounds, but that’s okay). Rinse well.

Cook leeks in a large pot of boiling salted water until meltingly tender, 12–15 minutes. Transfer to a bowl of ice water to stop the cooking. Drain and pat dry.

Whisk garlic, vinegar, and Dijon and whole grain mustards in a small bowl; season vinaigrette with salt.

Arrange leeks on a platter; drizzle with vinaigrette and let sit at least 10 minutes (the longer, the better).

Just before serving, scatter walnut mixture over leeks and top with tarragon and lots of pepper.

Do Ahead: Leeks (without walnut mixture and tarragon) can be marinated 1 day ahead. Cover and chill. Bring to room temperature before serving.