01 - Preheat grill or grill pan to medium-high heat.
02 - Brush chicken breasts with olive oil and season generously with salt and black pepper on both sides.
03 - Grill chicken for 6-7 minutes per side until fully cooked and juices run clear when pierced. Transfer to plate and let rest 5 minutes before slicing 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 as needed.
05 - Add chopped romaine lettuce to the bowl with dressing and toss until lightly coated.
06 - Add croutons and half of the shaved Parmesan cheese, then toss gently to combine.
07 - Divide salad among four plates. Top each serving with sliced grilled chicken and remaining Parmesan cheese.
08 - Serve immediately while chicken and croutons are still warm.