Can a composer build an orchestra piece out of one note's overtones?

Yes: that is what spectral music does. It uses the acoustic properties of sound, its spectrum of overtones, as a basis for composition. The approach originated in France in the early 1970s, developed primarily at IRCAM in Paris by composers such as Gérard Grisey and Tristan Murail.

A defining demonstration is Grisey's Partiels (1975). Its opening is derived from an electronic sonogram analysis of the attack of a low E on a trombone. The orchestra then plays that sound back to you, assigning different instruments to each harmonic partial. Lower partials enter one after another, the fifth and ninth louder than the lower ones, and each is rounded to the nearest quarter tone, so eighteen instruments share out the spectrum of a single trombone note.

Spectral music was partly a reaction against the pitch-focused aesthetics of serialism. Its pioneers say it is not a style: Murail called it an aesthetic, an attitude, and several major composers associated with it consider the very word reductive. Whatever the label, it reverses an old assumption. Instead of building sounds from notes, it builds notes out of the insides of sound.