Core the tomatoes and halve them crosswise, then squeeze the juices and seeds into a large bowl. Whisk in 1 tablespoon olive oil and season with salt and pepper. Add the arugula but don't toss; set aside. Dice the tomatoes and toss in a separate bowl with the olives and rosemary.
Preheat a grill to medium high. Brush the onion rounds with olive oil and season with salt. Grill until soft, 3 to 4 minutes per side. Transfer to a plate and separate the rings. Reduce the grill heat to medium.
Brush both sides of the pitas with olive oil and grill until marked, 2 to 3 minutes per side. Top with some of the tomato-olive mixture, ricotta, mozzarella and onion. Cover and grill until the cheese melts, 2 to 3 minutes.
Toss the arugula with the dressing and pile on top of the pitas. Season with salt and the red pepper flakes and drizzle with olive oil.
Photograph by Antonis Achilleos
Recipe courtesy of Food Network Magazine