To be extremely embarrassed. Usually the embarassment is the result of one's own actions. Not sure of the origin, but it may come from when clowns are at a circus and they have eggs thrown at them because of their goofy acts or when actors had eggs thrown at them performing in plays a couple of centuries ago.

Referring to the heavy unfashionable winter jacket and patterned mittens Sanders wore to the inauguration of Joe Biden.

Meaning - To be embarrassed.
Origin - Yellow egg shows up vividly well on your face if you don't wipe your mouth after eating, and, unless you're really thick-skinned, that's embarrassing.

Egg on your face is a clear synonym for having ejaculate on one's face.