Classical piano solo—Your one favorite performance and recording

An impossible choice, but I would nominate Brendel’s Phillips recording of Schubert’s last 3 sonatas

To be fair, I think there may be some confusion here with terminology.  There is a Genre (I hate that term, but it’s useful here) of Classical Music, and then within that Genre there is the Classical Period, the dates of which were supplied by another poster.

  I took the OP to mean any recording within the genre, not the historical period.  The recording that the OP cites is of Baroque Music, albeit on a more modern instrument than the Composer would have known.

  The Vikingur Olaffson Debussy/Ravel disc cited previously is another stunner

I remember being surprised when I read a book on Music Appreciation that lumped Beethoven and Schubert together as belonging to the Classical Period but being transitional figures, paving the way for early Romanticism