On the other end of the scale from the famous British chip shop is The Witchery by the Castle. This is somewhere that I always, without fail, recommend to friends visiting Edinburgh if they want one really flash meal. There are other restaurants in Edinburgh that are rated higher (such as Number One at the Balmoral or Martin Wishart at the shore, which both have Michelin stars), but restaurants such as those are a whole other level of 'fine dining' and price.
The food at The Witchery is top class, with a wonderful setting, but does not have the stuffy formal setting some of the others may. And the price is much better. (So if you are looking to splash out on a £100 meal, rather than a £400 meal.)
Aside from the wonderful food and agreeable service, for me the setting of The Witchery is its best selling point. I love it as a local, but it is its appeal to those visiting the city that makes it really great, as it so nicely sums up the vibe of Scotland's capital.
The walk along the cobblestone Royal Mile leads you up the winding and ancient street towards the castle at the top. And just before you get there, in a little nook just off to your left sits The Witchery. It would be easy to miss if you weren't looking for it. From the street you can look down into the windows of it and see the giant dripping candles along the windows, giving you an idea of the setting for this fantastic restaurant.
Once inside, it is a very nice, low lit setting. It's got a quiet and relaxed atmosphere, and the staff are very friendly and helpful. Candles flicker off the dark walls and amazingly painted and gilded ceilings. The entire restaurant is filled with tapestries and mirrors and paintings depicting the old Scottish links with France in the Auld Alliance. It is an amazing place just to see, and the food in such a setting does not disappoint.
They have a frequently updated menu (at an average of £30 per main course) full of local Scottish produce including the outstanding Aberdeen Angus beef and game. My first visit to The Witchery involved an amazing wild rabbit main course, while my friend had the Aberdeen Angus. And I must tell you, as a Texan, I am almost ashamed to concede that Angus was quite possibly—to this day—the best steak I have ever tasted. It was of incredible quality and cooked to perfection. And though it wasn't a 16-oz. T-bone, it was not a tiny portion, either. This is not a restaurant where you will leave hungry, if even the portions aren't huge.
They have an award-winning wine cellar including the Wine Spectator Best Award of Excellence in 2004.
I highly recommend you have a look at their website before heading in this direction. It's very comprehensive and well done. A top class restaurant.