Scrape gills from mushroom caps, if desired. Drizzle mushrooms with oil. Sprinkle salt, pepper and crushed Italian seasoning.
On charcoal grill, cook mushrooms on rack of uncovered grill directly over medium coals for 6 to 8 minutes, turning once halfway through cooking. Top each mushroom with a cheese slice. Place rolls, split sides down, on grill rack. Grill 2 minutes more, until cheese is melted, mushrooms are tender, and rolls are toasted.
Serve mushrooms on rolls. Pass mayonnaise, sweet pepper pieces, and basil leaves. Makes 4 servings.