Cook the pasta: Bring a large pot of salted water to a boil and cook the pasta according to the package directions until al dente. Drain.
Toss with oil: While the pasta is still warm, toss it with 2 tablespoons of the olive oil. Spread it on a baking sheet or large plate and let it cool to room temperature, about 15 minutes.
Make the dressing: In a medium bowl, whisk together the mayonnaise, parmesan, lemon juice, garlic, Worcestershire sauce, Dijon mustard, the remaining 2 tablespoons olive oil, salt, and pepper until smooth.
Combine: In a large bowl, combine the cooled pasta and shredded chicken. Pour the dressing over the top and toss until everything is evenly coated.
Chill: Cover and refrigerate for at least 30 minutes so the flavors have time to settle. Keep the romaine separate for now so it stays crisp.
Serve: Just before serving, fold in the chopped romaine, then top with croutons and extra parmesan.
Notes
Toss the warm pasta with olive oil before it cools, that's what keeps the noodles from clumping once the creamy dressing goes on. Fold the romaine in just before serving so it stays crisp. Total time includes 30 minutes of chill time.