Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, about 8 to 10 minutes. Drain and rinse under cold running water immediately to stop cooking and cool the pasta. Transfer to a large bowl.

Season the chicken breasts with salt, pepper, and a little olive oil. Grill over medium-high heat or pan-sear for about 6 to 8 minutes per side, until the internal temperature reaches 165°F. Let rest for 5 minutes, then slice or cube.

Wash and chop the romaine into medium-sized pieces so it mixes easily with the pasta.

Make the dressing by whisking together 1/2 cup mayonnaise, 1 tablespoon lemon juice, minced garlic, anchovy paste (if using), 1 teaspoon Dijon mustard, 1 teaspoon Worcestershire sauce, and 1/4 cup grated Parmesan until smooth. Taste and adjust lemon, salt, or pepper.

In the large bowl with cooled pasta, add the sliced chicken, romaine, croutons, and 1/2 cup Parmesan. Add enough dressing to coat everything evenly and toss gently. Season with salt and freshly ground black pepper to taste.

Refrigerate for at least 30 minutes to let the flavors meld. Before serving, garnish with extra Parmesan, a grind of black pepper, and lemon wedges if desired.