Flapjacks/thin pancakes on Westside?
Kate Mantilini's in Beverly Hills and The Fountain Coffee shop inside the Pink Palace (aka "The Beverly Hills Hotel" on Sunset Blvd.) both come to mind. Not quite traditional west side, but if you can make it between 11 AM and 3 PM (or it might be 2 PM) and not on Sunday when they are closed, Musso and Frank's flannel cakes are thin and excellent...
Beverly Hills Hotel
9641 W Sunset Blvd, Beverly Hills, CA 90210
Musso & Frank Grill
6667 Hollywood Blvd, Los Angeles, CA 90028
Kate Mantilini Restaurants
9101 Wilshire Blvd, Beverly Hills, CA 90210