Preheat oven to 375°. Put mushroom caps in a greased 15x10x1 in. baking pan. Bake 10 minutes. Separately, mix mayonnaise, Asiago cheese, basil and pepper in a food processor and process the mix until blended.
Drain the juice from the mushrooms. Fill each with 1 rounded teaspoon mayonnaise mixture and then top each mushroom with one half of cherry tomato.