Preheat an outdoor grill to medium heat.
Whisk the vinegar with the anchovy, if using, mustard, salt, and pepper in a small bowl. Gradually whisk in the oil, starting with a few drops and then adding the rest in a steady stream to make a smooth, slightly thick vinaigrette. Whisk in the parsley and tarragon and set aside.
Lay the romaine, mushrooms, and tomatoes out on a sheet pan and lightly brush them with olive oil and season with salt and pepper. Place the mushrooms on the grill smooth side down. Grill until juices collect in the cap and mushrooms soften, about 3 minutes. Turn them over and cook until slightly charred around the edges and the center is very tender, about 3 minutes more. Put the tomatoes on the grill skin side down and grill until juicy and charred, about 6 minutes. Grill the romaine, turning to slightly char all sides, until the core end is tender, about 5 minutes.
Divide the wedges of romaine, mushroom caps, and tomatoes among 4 plates. Drizzle the vinaigrette over the salads and serve.
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