Found a really good place today that has Cuban sandwiches. It's at 229 Main Street in El Segundo, called Havana Sandwich Shop.
They don't have waitress service, it's one of those place your orders and pay, and they bring the food out to you. In addition to offering the usual garlic chicken and pork entrees with black beans, rice and plantains, they have terrific Cuban sandwiches, something that's a little harder to find in LA.
They have a pretty flexible sandwich menu, which basically means you can have a classic Cuban with the ham, shredded pork, cheese, lettuce and pickles, or vary it by substituting/adding roast beef, salami and a few other items to suit your taste. It comes on a thin, pressed roll, and the whole mess is toasted together into a delicious oozing sandwich.
The one time I had a Cuban sandwich in Miami it was served on white bread, so I'm not totally sure which is the authentic method, but this sandwich was great anyway. I've tried a few other Cubans around LA (including a very bad experience at El Rincon (?) on Sepulveda in Culver City, and this is far tastier.