For each sandwich, top a slice of bread with a thin layer of cheese, red peppers, basil leaves, and more cheese. Top with a second slice of bread. Brush both sides of the sandwich lightly with olive oil, then push two or three toothpicks all the way through the sandwich to secure.
Lay sandwiches over medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); close lid. Cook, turning to prevent bread from scorching (remove toothpicks when turning, if they catch on the grill), until sandwiches are well browned on both sides, with some darker grill marks in spots, and cheese is melted, 8 to 10 minutes total. Serve hot, while cheese is still melted.