Arrange the pitas cut side up on a work surface. Cover 4 of the pita halves with the provolone. Arrange the ham, pepperoni, olives and artichokes, each on its own quadrant of the 4 panini halves. Top with the basil leaves and mozzarella and close the pitas. Brush the tops with 1 teaspoon of the olive oil.
Heat 1 teaspoon of the remaining olive oil in each of 2 large skillets. Add 2 pita panini to each and press down with a smaller skillet. Cook over moderate heat, turning once, until the cheese is melted and the panini are crisp, about 10 minutes. Cut into quarters and serve.