HAVANA, Sept 15 — Using a Soviet-era planer, Rene Reyes smoothens the panel of an old cedar door that he will transform into a humidor — an ornate, handmade box used to preserve Cuba’s famous cigars.

In a corner of the workshop lies a hodge-podge of discarded wooden beams, doors, and furniture, all used to make the cigar boxes, which can sell for thousands of dollars and are prized by wealthy ...