What's the name of this article of clothing?

A headress, often used with some styles of nuns’ habits, in which the head and neck are entirely enclosed leaving the face framed in an oval?

It’s a wimple.

I thought he was the old dude who was pimping toilet paper.

Wimple

That’s Mr Whipple (Mr. Whipple - Wikipedia)… ahhh, I saw what you did there.

That doesn’t seem to quite be it; what I’m thinking of makes the face look like they are looking through a cutout. ETA: unless what I’m thinking of is a sub-variety of wimple where the “framing” of the face is much tighter than some.

wimples can range from very tight to very loose.
http://www.bohemian.com/bohemian/02.20.08/stage-0808.html

http://lynnehurstinvites.com/edyth/notes/veil.html

It could also be a barbette

http://www.rameset.com/images/13thCenturyToque%20w-barbette.jpg

Whipple? Wasn’t he in that PBS series about The Bailey?

I think what you mean is still a wimple, but the way you wear it really makes a difference. For instance:

http://www.flickr.com/photos/borichar/3390167976/

Wimple, it used to be issued to female soldiers with their raincoat.

Ok, wimple it seems to be. Thanks all!

Did you hear about the nun who left her headdress at the convent when she got her new haircut? She went out not with a wimple, but some bangs.

You can get away with that once or twice, but don’t get into the habit.

A coif. The wimple went over the coif.
Nuns seem to have competed as to who could wear the most layers of clothing.

Unless you’re Robert Browning, in which case the word is “twat”

http://itre.cis.upenn.edu/~myl/languagelog/archives/001814.html