Bring a large pot of salted water to a boil. Add rotini and cook according to package directions until al dente. Drain and rinse under cold water to stop cooking. Toss with a drizzle of olive oil to prevent sticking, then set aside.
Preheat a grill pan or outdoor grill to medium-high. Rub chicken breasts with olive oil, garlic powder, salt, and pepper. Grill 6-7 minutes per side, or until internal temperature reaches 165°F (74°C). Let rest 5 minutes, then slice into thin strips.
In a medium bowl, whisk together mayonnaise, lemon juice, Dijon mustard, minced garlic, anchovy paste, and grated Parmesan. Add water a tablespoon at a time until dressing is pourable but still creamy. Season with salt and pepper.
In a large serving bowl, combine cooked pasta, chopped romaine, tomatoes, and shaved Parmesan. Add sliced chicken on top. Drizzle with dressing and toss gently to coat. Sprinkle croutons over the top just before serving to keep them crunchy.
Top with extra shaved Parmesan, a crack of black pepper, and serve with lemon wedges. Enjoy immediately or refrigerate for 30 minutes to let flavors combine.