What does "face as sharp as a pen" mean?

The phrase appears to be a reference to the controversial phrase from Shakespeare's Henry V (Act 2 Scene 3)

for his nose was sharp as a pen, and a'babbled of green fields

later

for his nose was sharp as a pen, and a table of green fields

google books

lrb.co

Reading the sources I quoted, one learns that this phrase baffled editors. Some thought it a mistake and corrected it. Some tried to figure out the real meaning behind it.

From the google book "Notes and Emendations to the Text of Shakespeare's Plays"

...writing tables were no doubt at that period often covered with green cloth; and it is to the sharpness of a pen, in relief of a table so covered, that Mrs. Quickly likens the dying wit of the philosopher...

Of course, all of this borders on speculation, but I'd say that your writer was trying to describe a similar situation.

Sparknotes translates this as gaunt.

So you could consider this to mean a gaunt face.