Get your grill up to about 550 to 600 degrees, dome temperature. Place the grid at normal cooking level if you have a Big Green Egg.
Remove the stalk carefully so as not to damage the mushroom cap or the gills. Place the caps in a large zip top bag, pour in enough Italian dressing to allow you to completely cover the caps when you seal up the bag, and then turn it over and over to coat the mushrooms. Let them marinate for 15 minutes.
Place the mushroom caps on the hot grid, gill side down. (This is the underside of the cap, so place them on the grill the same way they grow out of the ground, gills down, rounded cap up.) Grill for 1 minute.
Remove the mushroom caps to a platter, gill side up. Spoon enough of the tomatoes onto each cap to loosely cover the gills, about 1 heaping tablespoon. Place a slice of cheese over the tomatoes. Return the caps to the grill, cheese side up. Close the cooker's lid and let the temperature drop to 400-500 degrees. Cook for about 2 minutes or until the cheese is nicely melted. Remove to a platter with a spatula and serve.