As the biggest fast-food chains have expanded menus to attract new customers, Chick-fil-A Inc. has gained ground with a different strategy: its focus on chicken sandwiches.

Since its founding in Atlanta in 1967 by entrepreneur Truett Cathy, Chick-fil-A’s signature item has been its breaded chicken-breast sandwich served with pickles on a buttered toasted bun. The company sells breakfast and salads, too, but its relatively simple menu stands out, compared with its fast-food rivals.

Now, five decades later, the closely held company this year is poised to become the third-biggest U.S. restaurant chain by sales behind McDonald’s Corp. and Starbucks Corp. , according to food-service consultancy Technomic Inc. The growth reflects Chick-fil-A’s expansion over the past decade, as well as consumers’ broader turn toward chicken.

“We’ve been pretty consistent in how, over multiple decades, we’re going to take it slow and steady,” Mark Moraitakis, a Chick-fil-A senior director, said about the company’s expansion. “It’s paid off for us.”

Sales from Chick-fil-A’s restaurants have tripled over the past decade, reaching $10.2 billion last year, Technomic said, even though the chain is closed on Sundays, a practice implemented by its conservative Christian founder.

Chick-fil-A, also known for its customer service, has nearly doubled its store count to about 2,400 restaurants since 2007. The company now has locations from Manhattan to Hollywood. Its growing presence will test whether consumers will stick with the chain as they have in the South, Chick-fil-A’s historic base.

“I don’t know if it will have such an abiding loyalty in other regions,” said Sara Senatore, a restaurant analyst at Sanford Bernstein.

The company has met resistance when moving into other areas, in part because of public reaction to comments made in 2012 by Chief Executive Dan Cathy, son of the company’s founder, regarding the CEO’s opposition to same-sex marriage. Advocates called for a boycott and organized protests at Chick-fil-A restaurants, while conservative customers expressed support.

This year, officials in San Antonio stopped the chain from opening a location in the city’s airport because of the views on sexual orientation. The concessions company at the airport serving Buffalo, N.Y., dropped Chick-fil-A as a potential tenant at that facility after facing protests over the same issue.

A spokeswoman said the company focuses on ensuring that all customers, employees, restaurant operators and other partners are treated with care and respect.

“It does look like most consumers, for the most part, are looking the other way,” said Jeff Farmer, a restaurant analyst at Gordon Haskett Research Advisors.

Each year since 2015, Chick-fil-A has been the top-rated fast-food restaurant on the American Customer Satisfaction Index, which considers factors such as staff courtesy and restaurant cleanliness.

Bradley Ackerman, a professional drummer in Nashville, said he likes Chick-fil-A’s consistent service. “The wait is never too long. My order is never wrong,” Mr. Ackerman said.

Younger diners have given Chick-fil-A high marks. In a survey this spring of teenagers, respondents listed Chick-fil-A as their favorite restaurant, beating out Starbucks for the top spot, according to investment bank Piper Jaffray Cos.

Helping Chick-fil-A is an increase in consumer interest toward chicken. Fast-food restaurants in the U.S. served more than 2.5 billion breaded-chicken sandwiches last year, according to market-research firm NPD Group Inc., up 17% over a decade. Burger servings slipped 5% in that period to 7.4 billion last year.

Among limited-service U.S. restaurants that mainly serve chicken, Chick-fil-A’s market share rose to 33% last year from 18% in 2009, while the market share of Yum Brands Inc.’s KFC chain fell to 15% from 29% in that time, according to data tracker Euromonitor. Chick-fil-A’s growth also has hampered burger sales, consultants and industry executives say.

Other chains are tapping into chicken’s rising popularity and tweaking their menus.

KFC is planning to open new U.S. restaurants, on a net basis, in 2019 for the first time in nearly 15 years. McDonald’s has simplified its menu, eliminating a line of customizable burgers in April to streamline operations. Wendy’s Co. recently bolstered its chicken-sandwich lineup.

Chick-fil-A has kept its menu relatively straightforward. The company has made an average of 12 permanent or limited-time additions to its menu each year since 2010, according to Datassential, a food-research firm. McDonald’s, by contrast, has made 49 in recent years, and Burger King, owned by Restaurant Brands International Inc., has made 37.

Atlanta-based Chick-fil-A Inc.—the corporate parent—reported revenue and rents of more than $3 billion last year, up 28% compared with 2016. Net earnings rose to $435 million from $371 million, according to franchise documents some states make available.

The company says its operators, most of whom run just one restaurant, help assure consistent food quality and operations.

“It’s almost a calling,” said Stephen Browning, who opened a Chick-fil-A in Pueblo, Colo., in 2014.

The average McDonald’s franchisee owns a half-dozen stores, a company executive said in March. Carrols Restaurant Group Inc. operates 1,010 Burger Kings in the U.S. Chris Finazzo, president of Burger King North America, said wider reach is a strength, not a weakness.