The piece
Beethoven titled both sonatas of Op. 27 Sonata quasi una fantasia, and this one, dedicated to Countess Giulietta Guicciardi, does begin like a slow improvisation rather than an allegro. The "Moonlight" nickname is not his. At the head of the movement he asks for the whole piece to be played delicatissimamente e senza sordino, that is, with the dampers raised: the pedal is part of the writing.
The right hand has two jobs: the unbroken triplets and, above them, the melody given to the fifth finger. It has to sing with the top of the hand and murmur with the rest; some gaps between the tune and the triplet reach fourteen semitones. Watch the dotted rhythm in the melody: the sixteenth falls just after the last triplet of the beat, never with it. The left hand places sustained octaves in the bass, 98 of them, without harshness. Change the pedal with each harmony to avoid a fog. Over five minutes of pianissimo, the stamina required is mostly in your listening.