Place the chicken breasts between 2 pieces of parchment or waxed paper and pound thin with a rolling pin. Put them in a shallow bowl; add enough olive oil to cover (1/2 to 3/4) cup), the juice of 1/2 lemon, and salt and pepper, and refrigerate for 20 to 30 minutes.

Place a large grill pan on 2 burners over medium-high heat, or preheat an outdoor gas or charcoal grill.

Stir together the tomatoes, arugula, onion, olives, 1/4 cup of olive oil, the juice of the remaining lemon half, and salt and pepper; set aside while you cook the chicken.

Fold a few paper towels into a thick square. Blot some oil on the paper towels and then carefully and quickly wipe the ridges of the gill pan or the hot grates of the grill to make a nonstick surface. Remove the chicken from the marinade and season with salt and pepper. Lay the chicken on the hot grill and cook 3 to 5 minutes on each side until charred and firm. Transfer to plates and spoon the tomato mixture over. Serve hot.