Prepare a charcoal or gas grill for high heat (450° to 550°; you can hold your hand 5 in. above cooking grate only 2 to 4 seconds). Coat chicken breasts with 1 tbsp. oil and cook, turning once, until no longer pink in the center and grill marks appear, about 7 minutes total. Let rest 10 minutes, then slice.
Meanwhile, in a small bowl, whisk lemon juice, 1/3 cup oil, oregano, and pepper; set aside. Bring a medium pot of salted water to a boil, drop in asparagus, and cook until bright green, about 2 minutes. Drain and rinse thoroughly with cold water.
In a medium bowl, combine chicken, cheese, asparagus, tomatoes, olives, and pita chips. Pour reserved dressing over mixture and toss gently to coat. Add arugula and toss once more just to combine.
Note: Nutritional analysis is per serving.