1. Heat grill pan to medium-high. Mist chicken breasts with olive oil spray, and season with 1/4 teaspoon each salt and pepper. Grill chicken until just cooked through, about 5–6 minutes per side. Place chicken on a cutting board and let rest about 5 minutes to allow juices to redistribute; cut into bite-size pieces.
2. Mix next 8 ingredients and remaining salt and pepper in a blender until well-combined and creamy, scraping down blender sides, as necessary. Add 1 tablespoon water to thin. Stir in grated Parmesan.
3. Toss lettuce, croutons, and dressing in a large bowl; divide among 4 plates. Arrange chicken over salads, sprinkling each evenly with shaved Parmesan.