Whisk together grated Parmesan, lemon juice, mayonnaise, garlic, Dijon, anchovy paste, and Worcestershire. Whisk in ⅓ cup oil until emulsified; season with salt and pepper. Toss romaine and croutons with dressing; garnish with shaved Parmesan.
Rub 2 skinless, boneless chicken breasts with 1 tbsp olive oil, season. Place pan over a medium heat for 1 min, until hot, but not smoking. Lay the chicken on the pan (it will sizzle if it’s hot enough) and leave for 4 mins.
Turn the chicken, then cook for 4 mins more. Check if it’s cooked by poking the tip of a knife into the thickest part; there should be no sign of pink and juices will run clear.
Add to bowl of salad and enjoy!