Classical piano solo—Your one favorite performance and recording

I can’t pick just one, but perhaps something by Yevgeny Sudbin on BIS. If you like Scarlatti on the piano, his is excellent (and I usually prefer harpsichord in this). If not, his Medtner and Rachmaninoff album also is excellent.

Classical music is loosely defined as being composed from about 1750 to 1820.

This includes that from the minds of Beethoven, Mozart and Haydn.

Rach was pretty recent, in the scheme of things.

JS Bach could be correctly categorized as baroque.  What was the question?

Here's two nice ones

As mike in nc said, I can't give you just one - I have too many to choose from, maybe far too many. But just to be a little bit obnoxious, you mentioned Gould and Bach. If you want to hear a modern recording of the Goldberg Variations by a young and excellent pianist, you should hear Beatrice Rana's version on Warner Classic's. My preferred version of all available to me. :-)

FWIW one CD that gets a lot of play time from me is a compilation of very assessible but more modern (20th century) 'classical' music. 'Michael Sheppard plays Rogers/Hough - Barber/Sheppard  Crumb  Corigliano  Wild  Bolcom. 

If you can drill down your request to either composers, composition, or performers I, and others I think, could give you many specific recommendations.

John O'Conor's traversal of the Beethoven sonatas on Telarc is very good.  As are plenty of other recordings.