India, Oct. 2 -- Few watchmakers embody the metiers d'art quite like Jaquet Droz. These are the decorative crafts that elevate watchmaking into art: miniature painting, engraving and mechanical automata.

Since 1738, when Pierre Jaquet-Droz astonished Europe with his singing bird clocks and androids, the manufacturer has been known for blending mechanical ingenuity with artistic expression. Now part of the Swatch Group, it remains one of the most exclusive names in haute horlogerie.

In recent years, Jaquet Droz has taken a radical new path. Boutiques were closed, traditional distribution ended, and production was cut from thousands of pieces to no more than a hundred unique creations a year. Prices start in the low six figures (about $11...