Frederic Cassidy, Dictionary of American Regional English (1985) spells the term with a hyphen, although it acknowledges at least one variant spelling:
chicken-fried steak n Also chicken-fry steak chiefly West A steak, usu an inexpensive cut, breaded and fried. [Examples from the 1960s and 1970s—including one unhyphenated instance of "Chicken Fry Steak"—omitted.]
Merriam-Webster's Eleventh Collegiate Dictionary (2003) insists on a single spelling, with hyphenation:
chicken-fried steak n (1952) : steak coated with batter, fried, and served with gravy.
The American Heritage Dictionary of the English Language, fifth edition (2010) views "chicken-fried" as an all-purpose modifier, not inextricably bound to "steak":
chicken-fried adj. Coated with butter and seasoned flour and fried: chicken-fried steak.
The idea that this compressed term conveys is "steak cooked in approximately the same manner as fried chicken"—not "steak prepared by a gallinaceous fry-cook." It is an extremely popular menu item in small-town eateries in Texas (where I grew up), although in my experience "chicken-fried bootleather" might in some cases be a more accurate term for it. Such restaurants tend not to seek out authoritative sources of spelling and punctuation for the items they offer before committing them to paper (and lamination), which probably contributes to the real-world variations one may encounter in the spelling and punctuation of "chicken-fried steak."
On a peripheral matter, I note that Merriam-Webster's first occurrence date of 1952 for the term is way off. Instances of "chicken-fried steak" appear in newspapers at least as early as June 1, 1924 (in Indianapolis, Indiana), October 21, 1926 (in Stephenville, Texas) and January 31, 1930 (in Eagle Rock [a neighborhood in Los Angeles], California). Instances of "chicken fried steak" appear as early as November 17, 1920 (in Breckenridge, Texas). Overall, Elephind newspaper database searches turn up 458 instances of "chicken-fried steak" and 5,920 instances of "chicken fried steak," suggesting that lots of folks feel no obligation to hyphenate the dang thing, regardless of what the dictionaries may say.