"Deadpan" is a strange word. How does a cooking container that's apparently given up its ghost translate to an expression that means "emotionless" or "straight-faced"?
It starts to make some sense once you learn that "pan" in "deadpan" means "face." So, "deadpan" literally means a dead or emotionless face. Think about the flat delivery and expressionless faces of comedians and actors known for dry-witted humor.
"Pan" goes all the way back to Old English and could refer to the skull, especially the upper part of the skull. That's how we get compounds like "brainpan." It's not until the early 20th century that "pan" comes to refer to the face or the mouth.
The Oxford English Dictionary and Green's Dictionary of Slang both put "deadpan" back to 1920. Here's an early example from Ring Lardner's 1921 novel The Big Town: "Before I could open my pan, she says, 'Alright, and tell her we can't come.'" In other words, before the narrator could open their mouth to say something, someone else started talking.
"Deadpan," in the sense of a serious or emotionless face, came along in the early 20th century. It first showed up in the world of theater, as seen in this quote from a 1915 edition of the St. Louis Star-Times: "He had a trick of what the actors call 'the deadpan.' He never cracked a smile and would be the last man you'd suspect was working a trick."
In the early 20th century "deadpan" was seen as a slangy term, but that's not really the case today. According to Google Books, there's a dramatic rise of the use of "deadpan" in written English. "By the 1980s you see a dramatic rise in written English of 'deadpan,' and it may [have increased] as this word stopped feeling slangy," said Professor Anne Curzan.
During her research of "deadpan," Professor Anne Curzan came upon another "pan" expression, which is "shut pan" or "shut your pan." Based on what we've already learned about "pan," you can probably guess that this means "shut your mouth." For more on that, including an interesting origin theory involving Abraham Lincoln, listen to the audio above.