Stir together mayonnaise, shallot, mustard, tomato paste, garlic powder, vinegar, salt, Italian seasoning, pepper, and 1 tablespoon sundried tomato oil in a large bowl until combined. Stir in Parmesan.
Drain and chop sundried tomatoes into small pieces; add to mayonnaise mixture. Add chicken, celery, and basil to mayonnaise mixture and stir until well combined.
Divide chicken mixture between 4 slices of bread. Top evenly with arugula and remaining 4 slices of bread.
