Here in LA, there are a few "fancy" places I can recommend. These are operations which, while in the city, there are only one of, and which have been around FOREVER, which kind of makes them "destination" stops.
1) The Dresden Room. East Hollywood, unchanged since the '60s, except the food, which has recently improved. Steaks, prime rib, lasagna, $$
2) Musso & Franks. Hollywood Blvd, since 1919. Steaks, lots of veggies, will never change a bit. Great bar. $$
3) Taylor's Prime Steaks. West of Downtown, I've seen John Cusack in here. Good steaks, baked potatoes, great burgers at lunch. Nice mushroom soup on Wednesdays. On 8th St. $$
4) Pacific Dining Car. Also since about 1920. 6th St West of DT. OPEN
24 HRS! Best breakfast in LA served all day. Serious steaks. $$-$$$$
5) Chez Jay. Santa Monica. Steaks, seafood. 1940's style. $$
6) The Dal-Ray. San Gabriel Valley. Steaks, etc. 1950's. $$
7) Tom Bergins. Fairfax south of Wilshire. Irish restaurant/pub.
Historic location. $-$$
Okay, when I say $$, I mean at least $15 entrees, $$$ means $25, and $$$$ means maybe a $30 steak.
Now, I know there will be other suggestions, but I would enjoy hearing some comments about mine. Oh, I almost left out El Cholo (1928 I think), on Western west of DT. There are two new ones in Santa Monica and Pasadena, but you want the original. Classic "Mexican Food", $$.
-Scott Lindgren firstname.lastname@example.org