Rinse and drain watercress; pluck off tender sprigs (you should have 1 1/2 to 2 cups). Chop enough of the remaining watercress (including stems) to make 1/4 cup; save leftovers for other uses or discard.
Trim off and discard any mushroom stems to make ends flush with interior of caps; rinse caps and pat dry. In a small bowl, mix olive oil and 1 tablespoon Worcestershire. Rub mixture evenly over smooth sides of mushroom caps. Sprinkle both sides lightly with salt and pepper. Lay caps, gill side down, on a rack in a 10- by 15-inch broiler or baking pan.
Broil about 4 inches from heat until juice starts to drip from mushroom caps, 4 to 5 minutes. Turn caps over and broil until juice in caps bubbles and mushrooms are flexible, 5 to 7 minutes longer.
Meanwhile, combine shallots, vinegar, and tarragon in an 8- to 10-inch frying pan; boil over high heat, stirring often, until all the liquid has evaporated, about 1 minute. In another small bowl, mix cornstarch with 2 tablespoons water until smooth; whisk in milk, mustard, and remaining 1 teaspoon Worcestershire. Add to shallot mixture and stir until boiling. Add salt and pepper to taste.
Transfer mushroom caps to dinner plates. Tuck watercress sprigs alongside caps. Spoon sauce equally over mushrooms and sprinkle with chopped watercress.