Coat the chicken in a little olive oil, salt, and pepper. Roast for 20 minutes in a 350F oven. Shred the chicken once cooled.
Cook the pasta per package instructions in salted water.
Combine the egg, lemon, garlic, mustard, anchovy paste, and salt/pepper in a food processor. Pulse a few times until smoothed.
With the food processor on low, slowly drizzle in the olive oil until the dressing becomes thickened.
While the pasta is still hot, add about half the dressing, the chicken, pine nuts, garlic, and basil. Toss in a bowl and allow to cool for about an hour to room temp. Then cover and cool in the fridge.
When ready to serve, coat the lettuce in a little bit more of the dressing. Add the lettuce, tomatoes, and shaved parmesan to the salad.