COPENHAGEN — When Peter Enevoldsen won a lucrative order for the precision tractor parts that his company, Sjorring Maskinfabrik, makes in northern Denmark, his eyes lit up. The contract was worth more than half a million euros — a boon for his profits.

There was just one hitch: He did not have enough employees for the job.

Delivery was delayed, by one month, then two, then three, as he searched for skilled welders to speed the work at the sprawling factory. But in Denmark’s fast-recovering economy, they were hard to come by.

As Europe rebounds from its economic malaise, Denmark is one of a few countries that can boast of nearing a golden era of full employment, meaning almost everyone who is able and willing to work has a job. But instead of being cheered, it is posing new challenges to the country’s recovery.

More than a third of companies in this industrial and technically advanced nation can no longer recruit enough skilled workers to fill posts. Vacancies abound for I.T. specialists, computer scientists, engineers and mechanics, as well as for electricians and carpenters. The wages needed to lure them are creeping up. Affected firms are scaling back production, turning down contracts and postponing expansion plans.