Brush the chicken on both sides with oil and season with salt and pepper, to taste. Grill until slightly charred on both sides and just cooked through, about 2 minutes per side. Put a few slices of cheese on each breast, close the cover and let it melt slightly, about 40 seconds. Brush the cut side of lemons with oil and grill, cut side down until golden brown and slightly charred, about 45 seconds.
Put the chicken on large serving plates, top each breast with 2 slices of prosciutto and some arugula. Squeeze the grilled lemon on top and drizzle with olive oil.