01 - Preheat grill or grill pan to medium-high heat.
02 - Brush chicken breasts with olive oil and season evenly with salt and black pepper.
03 - Grill chicken for 6 to 7 minutes per side until fully cooked and juices run clear. Allow to rest for 5 minutes, then slice thinly.
04 - In a large bowl, whisk together mayonnaise, grated Parmesan, lemon juice, Dijon mustard, Worcestershire sauce, anchovy paste, minced garlic, and black pepper until smooth. Adjust seasoning to taste.
05 - Add chopped romaine lettuce to the bowl with dressing and toss until lightly coated.
06 - Add croutons and half the shaved Parmesan cheese, then toss gently to combine.
07 - Divide salad among serving plates. Top each portion with sliced grilled chicken and remaining Parmesan cheese.
08 - Serve immediately while fresh and crisp.