The way chords are voiced are important to Evans's style. He was one of the first to think in terms of rootless voicings, upper structures, alternating close seconds and thirds, harmonic planing using 4ths etc..all these devices came from his superior knowledge of 19th century piano Impressionism.
Nowadays of course this is as much of a modern jazz cliche as anything else but it was revolutionary at the time and paved the way for Keith Jarrett and the style of music known simply as 'ECM' (one if the few occasions a record label gives its name to a type of music).

In many ways Evans and the Guaraldi example are the opposite of thinking in terms of one chord at a time. The affecting example Mac posted to above is a classic case of what is essentially 're-composition'or 're-harmony'. Structural awareness in that sense is the ability to avoid a literal interpretation of the original. That involves knowing the essential features of the melody, where the strong points of harmonic resolution are and then being able to take a different route to that resolution from the original.

Alan