What is your formal pick for DC?
Ok, out of these restaurants, which are elegant and formal restaurants for teenagers?
Taberna del Alabardero
Yabba Pot (Virgin Islands/Caribbean-influenced vegetarian/vegan - not gonna find that in too many places!)
Ras Doobie (good Jamaican, and Doobie is just a really wonderful, gentle soul)
Martick's (quirky "French influenced" in the most anonymous location imaginable)
Matthew's (unique style of pizza)
Joe Squared (unique style of pizza - sourdough crust!)
Let me know!
My guess is you are looking for something that is not super expensive or formal, but still is elegant. I have two recommendations: Restaurant Eve (in Old Town Alexandria, not in DC) for quiet, classic. Zaytinya for chic, bustling. No reservations after 6:30pm at Zaytinya, so you might have to wait for a table if you plan on eating later than that.
Mansion on O, because it's like a playhouse.
701 because it is hushed and plush, has a piano, and has a wide menu that even includes mini-burgers.
Dino, because it has lovely service and an accessible menu.
All three will make 11 and 14 year olds feel very grown up and still have foods that they will easily recognize and enjoy in very pretty settings.
From the list you provided, I'm inclined to select the Palm if the kids like steak or lobster, because the kids will be entertained by all the caricatures on the wall and might have a pretty good chance of seeing a real live celebrity, too. Plus, the steak and lobster are as good as you'll find anywhere.
The Palm has an unfortunate tendency to fawn over its regulars who spend $1000s each month on fat expense accounts. Watching the staff lavish attention on celebrities won't make up for being ignored if the kids aren't treated as special guests too.
There are other places with great food that may treat them better.
I would recommend Dino [Italian], Zengo [Latin Asian fusion], Etete [Ethiopian]or 2 Amy's [Italian]. None are formal but all, I think, would be good spots for the teens--intersting and fun food. Mnay of the places you inquire about, e.g., 701, Obelisk, Palm, Vidalia, Restaurant Eve, Marcel's, La Paradou, are top restaurants but are not appropriate for most teens nor do I think that most 11-year olds would appreciate the food or service at these places.
re: Dakota Guy
I think you're selling kids short to relegate them to places that you think are "fun" just because they can eat unfamiliar Ethiopian dishes with their hands or mingle with the other kids over yet another pizza at 2 Amys. The OP asked about some sophisticated places for some kids he knows that might well enjoy a taste of the adult world and be well suited to an evening there.
The last time I ate at Citronelle, there were two boys about that age at a nearby table in coats and ties, with lovely manners, being hovered over by the adoring staff. They were ordering off the regular menu and seemed to be enjoying it thoroughly.
My own children went with us from an early age to good restaurants here and abroad and not only behaved well and didn't expect a "kiddie menu," but learned to really enjoy fine dining. By the time they were college-age, my visits to NY usually included a slice at lunch and dinner reservations, made by them for us and my AMEX, at places like Nobu.
Children love to be treated well and will live up to expectations.They see on TV and in movies how "fancy" places are supposed to treat patrons. They'll appreciate being on the receiving end.
I just wouldn't take them to the sorts of restaurants that are sometimes snotty to diners of whatever age if they aren't regulars or apparently wealthy.
Out of these I might suggest Restaurant Eve. The servers at Eve are so personable that I'd imagine them to be wonderful with kids. One thing I might suggest- when you make the reservations I might make a mention to the host that you are going to be bringing kids and wish to be with a server who will be good with them, but still formal.
Boys or girls? If you are willing to spend the money, Citronelle, Marcel's or Taberna del Alabardero. The first would be exceptionally good and the teens would get a real kick out of the dessert "breakfast." Taberna or Marcel's would be good because ithey are so "Old World" and the food is excellent, and the setting very formal. Dino would be fun, but is not "elegant" per se, but the food is good and my teenager and his cousin have loved it every time we have gone.