Trim mushroom stems flush with caps; save stems for other uses. Quickly rinse mushroom caps and drain on towels, smooth side up.
Place caps, gills down, in a 10- by 15-inch pan. Broil about 4 inches from heat for 5 minutes (about 4 minutes in a convection oven), then turn over and broil until juice in caps bubbles and mushrooms are flexible, 6 to 7 minutes more (about 6 minutes more in convection oven).
Meanwhile, cut cherry tomatoes in half and discard tough arugula stems.
Mix mayonnaise, mustard, and vinegar.
Cut each mushroom cap into thin, slanting slices. Divide arugula among plates. Top arugula equally with mushroom slices (and any juices) and cherry tomatoes. Spoon mayonnaise mixture onto salads. Add salt and pepper to taste.