The title of your post asks a question that is as broad and fundamental as wondering "how do I play the piano?", or "how do I make music?" This degree of puzzlement merits an essay in response, so I am going to give you one.
First of all, the way you write about, for example, Mozart sonatas, reveals a certain naivety about the interpretational demands of technically simpler repertoire, and about your concept of interpretation itself. I think traveling much deeper into the music you already play is going to be key to your development, whether in tackling "simple" or "advanced" repertoire.
Even concert pianists who play the most demanding romantic pieces by Liszt or Rachmaninoff, regard Mozart sonatas as incredibly challenging and elusive on an interpretational level. In fact, the core virtuosic romantic repertoire is often more approachable, because its sentiments and gestures are worn "on the sleeve", and there is not such an extreme sense of exposure.
But let's back up a bit - what do you think interpretation actually *is*?
"Sure I can phrase at an intermediate level..." you say, as if this means you have mastered the soul and wit of the Mozart sonatas.
Sure you can phrase... but what about building phrases of phrases of phrases, to create a sustained narrative structure that tells a story full of questions, surprises, human characters and emotions? What about the nuanced balancing of the inner voices of chords, or of one voice against another, to paint shades of hope against darkness, joy against vulnerability? And doing this not just with minutely contrasted dynamics, but using subtle shifts of articulation and rubato? And as for rubato, what about judging the micro-temporal adjustments necessary to make cadences dovetail just so? What about capturing the essence of a run or flourish by ascending to scarcely believable peaks of smoothness and evenness and gradations of touch? What about the spiritual quality of the music? What about a million other barely perceptible things that make all the difference?
I have to laugh, really, when you say that playing Mozart sonatas at this "advanced level" you have reached where you can "learn anything you want to play" would be "beating around the bush". (I am not laughing at you, because as a kid I reached the same false conclusions myself.) Might I gently suggest that a serious classical musician would find such statements absurd. After all, some of the most formidable technicians of our age devote years to contemplating Mozart sonatas. And have you listened to the recordings by Brendel, Pires, Uchida? Does your playing of the sonatas yet rank amongst these glories?
Once you have reconsidered all these things, and updated your concepts of technical proficiency to match the demands of expressing a new level of detail, fluency and personality, I don't think you will be asking such generalized questions anymore... instead of "how do I interpret piano music?", it will be "in this specific modulation, could I let the melody linger very slightly, with a gentle diminuendo, to increase a sense of unfulfillable longing?"... or "exactly how should I stagger the loudness and staccatissimo of these chords, to convey a certain dismissive brutality?"... etc. etc.
By all means start tackling more virtuosic pieces. But it sounds like you also need to start tackling K545 again, with a view to communicating something vital.
And I don't mean to encourage this view of a dichotomy between technique and artistry. The idea that interpretation can just be tacked onto a piece at the end of a course of technical training is a mistake that will lead you to dry and fatally uninteresting pianism. Your interpretative ability defines you as a musician, and you should have developed it alongside everything else; or rather, it should have become in some way unified with your technical intentions.
As I wrote recently wrote to another forum member: "Of course, the more advanced you are, the more generally fluent your playing; but even in the simplest piece, the serious pianist will always be able to find something to struggle with to the point of exhaustion."
I am not saying that a great concert pianist would have any particular technical problem with anything in, say, the Anna Magdalena notebook; yet if that pianist were going to seriously perform or record the notebook, be assured that he or she would spend several days peering into every nook and cranny of those pieces, turning them over and over in their minds, seeking, searching for an expressive vision that will capture hearts and minds. This in turn will bring them to some limits of their technical ability - how quietly and absolutely evenly they can give shape to a particular passage, how elegantly and wittily they can turn out of a trill... The interpretative and technical requirements are of the same nature, and, in the end, pianism is a oneness. If you can truly interpret the Mozart sonatas, the door to everything else will be wide open.
Topic: How do I interpret piano music?