Make dressing: In a small bowl whisk together yolks, vinegar, and mustard and add oil in a stream, whisking constantly until dressing is emulsified. Stir in egg whites, shallots, tarragon, and salt and pepper, to taste.
Thinly slice spinach leaves and spread on a platter. Cut mushrooms into very thin slices and mound on spinach. Spoon dressing over salad.
