“We went to the Middle East because there was wealth, there was an ambition and a market that was ambitious for higher-tier brands, and, yeah, there was a lot of youth,” he said. “Our concept was simple: We’ll go to Dubai. We’ll set up an amazing store in a good location — and we kind of saw that as a permanent sort of trade exhibition center for BurgerFuel — and we’ll see if we can attract interest globally from there.”

The Middle East has been a standout. Sales were up 95 percent to 16.7 million New Zealand dollars in the region as the company added five stores there in the year that ended in March 2013. The only other outlet outside New Zealand is in Sydney, Australia.

The focus on the Middle East has also led to projects in unexpected areas. BurgerFuel has opened a store in Iraqi Kurdistan, and a spokesman for the company said the first of three stores in Egypt was scheduled to open this weekend. A store in Kuwait is also expected to open soon.

The United States presents its own challenges for BurgerFuel. The fast-food business is already crowded, with 256,000 outlets and $204 billion in sales, according to Euromonitor International figures from 2012, the most recent data available.

Elizabeth Friend, a senior research analyst at Euromonitor International, said there was still some room in the upscale hamburger industry, which BurgerFuel is aiming at. But she said the company would have to distinguish itself from brands like Five Guys, Epic Burger, Umami Burger, Shake Shack and In-N-Out Burger.

“The better burger segment in the U.S. is highly saturated in terms of concepts, but not necessarily in terms of actual outlets,” Ms. Friend said. “The novelty of the specific ‘better burger’ model has worn off to the point where this positioning alone is no longer enough to set a brand apart. BurgerFuel will have to offer something truly better if they want to survive.”

Mr. Roberts pointed to Radio BurgerFuel, a company station that broadcasts to its stores, as a point of differentiation.