There is some debate about when the first “hamburger restaurant”—not merely a hamburger stand—opened for business. Louis’ Lunch of New Haven, Connecticut, claims to have invented the hamburger and to be the oldest continuously operating hamburger restaurant. However, Louis’ Lunch began as a lunch wagon in 1895 and didn’t become a restaurant until about 1917. Also, hamburgers weren’t served there until 1900, and some hamburger scholars insist that Louis’ Lunch serves a hamburger sandwich (on slices on bread), and not a hamburger (on a bun). White Castle is a hamburger restaurant chain that began in 1921.
 
The term “hamburger restaurant” was cited in 1904 in a newspaper from Fort Wayne, Indiana.
 
   
Wikipedia: Louis’ Lunch
Louis’ Lunch in New Haven, Connecticut, advertises itself as the first restaurant to serve hamburgers and as being the oldest hamburger restaurant still operating in the U.S. Opened as a small lunch wagon in 1895, Louis’ Lunch was also one of the first places in the U.S. to serve steak sandwiches.
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According to family legend, one day in 1900 a local businessman dashed into the small New Haven lunch wagon and pleaded for a lunch to go. Louis Lassen, the establishment’s owner, Louis placed his own blend of ground steak trimmings between two slices of toast and sent the gentleman on his way, so the story goes, with America’s alleged first hamburger being served.
 
In 1907, Lassen moved the business to Temple and George Streets. After a decade there, he left his lunch wagon for a square-shaped little brick building that had once been a tannery.
 
Wikipedia: White Castle (restaurant)
White Castle is an American regional fast food hamburger restaurant chain located in the Midwestern and Southern United States as well as in the New York metropolitan area, generally credited as the first fast food chain in the US. It is known for its small, square hamburgers.
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History
White Castle was founded in 1921 in Wichita, Kansas. The original location was the NW corner of First and Main. Cook Walt A. Anderson partnered with insurance man Edgar Waldo “Billy” A. Ingram to make White Castle into a chain of restaurants and market White Castle.
 
2 September 1904, Fort Wayne (IN) Evening Sentinel, pg. 10, col. 4:
HAMBURGER RESTAURANT.
Stop at the Hamburger restaurant, 1112 Calhoun street, and get a hot Hamburger for 5 cents.