The book American Slang gives the following entry for pan:
- The face. MUG. Too great for them to keep their pans shut. (1923+)
This seems to be when pan first entered American slang. From there, it went on to form words like deadpan (recorded in 1927). Only one site gave a hint to the actual origins, though. In this question about the origins of "deadpan", it is written that:
The key to "deadpan" is the use of "pan" as theatrical slang for "the face" (reflecting the use of "pan" to mean "skull," found as early as 1330).
Going off of this, there seems to be some support. The Word Origins page for skull says:
The Old English word for ‘skull’ was hēafodpanne, literally ‘head-pan’. It has never been firmly established where its Middle English replacement skull came from, but is seems more than likely that it was borrowed from a Scandinavian language (Swedish and Norwegian have skalle ‘skull’).
So the old word for skull gave pan, which was adopted to mean face. From there, pan entered American slang in the early 1920s.