The crescents themselves were most likely turned with the piece mounted eccentrically to cut the inside, and carved to create the gap between the cusps.
Two other questions are at work here:
1. How did Jacques build it?
2. How would YOU build it?
Claude posted this on a few other forums, where he said the surface was achieved by wire brushing. Other than that, question no. 1 should be directed to Jacques, perhaps via Claude. (Claude is a member here, too.)
For question no. 2, speculation can run wild. But here's my take on it:
Rigging of constructions like this is best done with use of disposable temporary supports, arranged so as not to interfere with threading the strings. For the attached whimsy (made many many years ago), I attached the straws to the faces of a cardboard cube with interlocking edges. After threading the strings, I just cut away the cardboard. I note that Jacques has an unidentified insert on the rims of the pieces. I think my temporary supports would be sheets of heavy cardboard or Masonite engaging the slots for the later inserts. The sheets would extend to a surrounding box to hold the two pieces in position. The box would have access holes to facilitate threading. Even with that, though, tensioning the strings could be an adventure: the six strings on a guitar need to be re-tuned as each string is tensioned; on this piece it would be even worse.
Claude probably used some props to support the piece for photography, and edited them out of the picture files. At least that's how I'd do it.
Joe