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.