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Nearly 14 years ago, on something of a lark, the restaurateur Danny Meyer opened a Chicago-style hot dog cart in Manhattan’s Madison Square Park, hoping to draw crowds to the park and give summer jobs to the staff at one of his nearby high-end restaurants.

That stand has morphed into Shake Shack, a burger-and-crinkle-fries empire with outposts in London, Dubai, Istanbul and Las Vegas. On Friday, it will begin trading on the New York Stock Exchange with a valuation of about $745 million, and will increase Mr. Meyer’s net worth by about $155 million.

Conceived as a homage to the friendly Midwestern fast-food joints of Mr. Meyer’s childhood, Shake Shack has become one of the most prominent purveyors of fast-casual food. That sector, dominated by the likes of Chipotle, has fundamentally reshaped the fast-food industry with its emphasis on using fresh ingredients. In short, Americans seem willing to pay more for fast food made better, so long as they are still served quickly.

The success of Mr. Meyer’s chain stands in stark contrast to McDonald’s, the global behemoth suffering from its worst slump in more than a decade. The golden-arched restaurant chain announced a change in leadership this week facing sagging sales and a flat stock price, as it struggles to adjust its well-worn menu for modern tastes.

Mr. Meyer, 56, and his team have had no such trouble. Shake Shack has resonated with consumers who grew up on fast food but are both wary and weary of it. Burgers have been enjoying a makeover that began in the late 1990s, as younger eaters have flocked to a new generation of burger chains like Shake Shack, Five Guys and Smashburger.

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Mr. Meyer’s chain is part of a new crop of fast-casual restaurants that promote the authenticity of ingredients. Many have since gone public, stirring up investors’ appetites: Shares in Zoës Kitchen have doubled from the chain’s public debut, while those in El Pollo Loco are up 76 percent. The shares of another chain, Habit Restaurants, have risen more than 80 percent since their November debut.

Yet the fast-casual dining sector has become crowded, with a host of new entrants in an already competitive restaurant business. Shake Shack has tripled its store count in just two years, with 63 branches, and now Mr. Meyer and his team must prove they can manage their chain’s explosive growth and weather the public’s fickle tastes.

Shake Shack is rooted in Mr. Meyer’s own culinary experiences. Its origins lie in St. Louis, where he grew up on straightforward food served with Midwestern friendliness at restaurants like the German restaurant Schneithorst’s and Steak ’n Shake, itself now a 500-restaurant chain.

He also came to love the frozen custard at Ted Drewes, which began selling Christmas trees and frozen custard in the 1930s. That restaurant introduced the concrete, a shake as thick as ice cream with a raft of mix-ins, in 1959; it is now a signature item at Shake Shack.

So integral is the frozen treat to the company’s identity that Mr. Meyer nearly named the hot dog cart “Custard’s First Stand.” He acknowledges in the stock sale’s prospectus that the name was “pretty bad.”

But Shake Shack also draws on the lessons Mr. Meyer has learned in his three decades as one of New York’s most successful restaurateurs. His career began in 1985 when, at age 27, he opened the Union Square Cafe as a kind of antithesis to New York restaurants of the time that cultivated exclusivity and excess. The restaurant’s mix of warm service, dishes made with produce bought at the nearby Greenmarket and top-notch food served more casually was groundbreaking.

Since then, hospitality has been the calling card for his empire, which has expanded to what are now fixtures of the New York dining scene: Gramercy Tavern, Blue Smoke and Maialino among them. (Those restaurants are part of a separate company, the Union Square Hospitality Group, that will remain privately held.) The restaurateur has even written a best-selling book, “Setting the Table,” a primer on customer service.

Even in its prospectus, Shake Shack refers to its customers as “guests.”

“The thing I learned growing up in St. Louis,” he told St. Louis magazine in 2007, “was the power of hospitality. The enormously warm feelings of loyalty that come from feeling welcome and being recognized and having the sense that the restaurant is happy to see you.”

That combination of quality ingredients and warm service has proved profitable, though the company is still a relative minnow. Shake Shack reported $5.4 million in net income in 2013 on $82.5 million in sales. Chipotle, by contrast, reported about $327 million in net income on $3.2 billion in sales in the same year.

Still, its success has helped bolster the fortunes of Shake Shack’s owners and close partners. Beyond Mr. Meyer, the top shareholder is Leonard Green & Partners, a Los Angeles private equity firm that invested in the company in 2012. The firm’s partner now on Shake Shack’s board, Jonathan D. Sokoloff, was introduced to Mr. Meyer through top executives at Whole Foods, in which Leonard Green once held a stake. They initially met over dinner at Gramercy Tavern.

Responsible for helping propel the growth of Shake Shack, Leonard Green’s holding in the company is now valued at roughly $193 million.

Shake Shack has even helped transform Pat LaFrieda, which manufactures the company’s secret burger blend, from a local artisanal butcher into a nationally lauded purveyor of quality beef.

But Shake Shack’s ambitious expansion plans — the chain plans to open at least 10 company-owned restaurants in the United States each fiscal year — may threaten the high level of hospitality the company is known for.

Mr. Meyer’s original network of restaurants was opened within a tight radius within New York City, so that the restaurateur could walk between them and ensure that each was up to par. That hasn’t been possible with Shake Shack for some time, and its increasingly far-flung locations risk eroding that quality of service.

Already, its Manhattan-based locations are more profitable than its other branches, reporting 31 percent operating profit margins compared with 21 percent for non-Manhattan restaurants.

Another potential problem is that the so-called “better burger” slice of the fast-casual market is getting crowded, according to Darren Tristano, an analyst at the research firm Technomic. While the market may grow from $3 billion to $5 billion, he argued, it won’t grow much more — and consumers may slowly lose their hunger for burgers.

“I wouldn’t necessarily call them unique or the best, but they are a very well-positioned burger concept with good service,” Mr. Tristano said. “They’re not that much different from regional players.”

But Shake Shack is wagering that hungry customers will beg to differ.

“When Shake Shack opened up a block from my house,” the chef Anthony Bourdain said in 2011, “I dropped to my knees and wept with gratitude.”

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