Origin of "egg on my face"

It often implies that you have made a serious mistake, but more strictly it indicates that something you have done (or some turn of events) has left you looking embarrassed or foolish:

Murray knows all about the Chilean 12th seed — he practised with him last week — and the British number one realises he could be left with egg on his face if he leaves Gonzalez an easy, mid-court ball.

Where it came from (source):

I know of two suggestions for where it came from. The late John Ciardi suggested an origin in the lower-class and more rowdy kind of theatrical performance, in which an incompetent actor would have been pelted with eggs and forced off the stage. The other is that it was a comment on a minor social gaffe at a meal, when poor manners or sloppy eating left egg around your mouth.

And also (source):

From the embarrassment suffered if the yellow yolk is on ones lips or beard after eating a soft boiled egg in one of those egg cups, a favorite breakfast of the upper crust... Yellow egg shows up especially well on a beard or mustache.

Etymonline suggests 1964 as the first recording and includes its meaning.

To have egg on (one's) face "be made to look foolish" is first recorded 1964.

But another website claims an 1941 source and includes two guesses at the phrase's origin with the second being more likely:

The late John Ciardi suggested an origin in the lower-class and more rowdy kind of theatrical performance, in which an incompetent actor would have been pelted with eggs and forced off the stage. The other is that it was a comment on a minor social gaffe at a meal, when poor manners or sloppy eating left egg around your mouth.

I found an antedating of the phrase. This clip is from a January 4, 1936 issue of The Spokane Daily Chronicle. In this article, two friends gossip about a third friend (Marnie) who came for a visit but rudely rushed off because she had double-booked appointments. The article is full of idioms and seems to make a point of including all the latest fashionable slang of the time. Interestingly, the 1941 reference mentioned by @MrHen appears to have been an AP story and was printed in this same newspaper.

Also of note, it appears Michael Quinion has updated his entry on this phrase, since this question was asked, with another plausible origin of the phrase:

Subscriber Cal Clifford put a possible new perspective on the expression by mentioning egg-sucking dogs: “Occasionally, a trusted, working farm dog would develop the bad habit of taking eggs from nests and eating them, turning himself from asset into liability.” I found several examples of the term, including these:

His chief business was the doing away with dogs of ill-repute in the country; vicious dogs, sheep-killing dogs, egg-sucking dogs, were committed to Alan’s dread custody, and often he would be seen leading off his wretched victims to his den in the woods, whence they never returned. Glengarry School Days by Ralph Connor, 1902.

He’s a miserable, fox-faced scoundrel, and I’ve no more use for him than I have for an egg-sucking dog. Nan Sherwood at Pine Camp, by Annie Roe Carr, about 1919.

So it’s just possible that the expression might be a figurative extension from that of a dog found with egg around its muzzle, mute evidence of depravity.

While we know the current meaning of the phrase, and have some clues as to when and where it came into use, I want to suggest that speculations about how the phrase arrived are very likely permanently unanswerable, as with so many graphic phrases.

We know from Michael Quinion's citation on WorldWideWords that the phrase was in use in 1941, apparently as US teenage slang. I suggest that there was a somebody who first used it in this sense, though we will probably never know who that somebody was. It might be the case that that somebody had a clear image in mind, and could in principle distinguish between the two origins suggested above. They might even have told someone which. If they didn't tell somebody, then the question is permanently moot and any number of people speculating on what was the phrases's origin will still not produce an answer.

My own guess is that the person could not have said which meaning they were thinking of: either, or both, or a generalised image of what a person looks like with egg dripping down their face. Like many graphic phrases and metaphors, it doesn't need a precise origin - indeed the assumption that there must be one implicitly says that there is no such thing as imagination.

I may of course be wrong in this case: they may have had a clear image in mind. But if they didn't tell anybody, we can't know it.