Rub chicken with oil and season with salt, pepper, basil and oregano. Grill or broil chicken until browned and no trace of pink remains in the center, 7-8 minutes per side.
Combine lettuce, dressing, Parmigiano Reggiano, and salt and pepper in a large bowl. Toss and divide among 4 plates. Cut chicken into 1/2-inch slices and fan over salad. Top with Parmesan curls. Serve immediately.