A “pizza face” (or “pizza-face,” “pizzaface”) is a person with face full of pimples. The allusion is to the cheese on the pizza, which bubbles up on the best pizzeria pizza. (Frozen pizza often doesn’t have the same cheese bubbles.) “Pizza face” is first cited in print from a teen slang article in 1964.
 
A pizza is usually round, so it’s possible to make a face with the ingredients. In December 1964, Jeno’s frozen pizza advertized how to make “pizza funny faces.”
 
     
Thesaurus.com
pizza-face
Main Entry: acne
Part of Speech: noun
Definition: blemishes
Synonyms: blackheads, pimples, pizza-face, pustules, rosacea, skin inflammation, whiteheads, zits
 
Urban Dictionary
pizza face
An offensive term telling people that they have bad acne
Girl with acne: Dorothy, you’re such a slut, you ugly bitch.
Dorothy: Get Neugtrogena, pizzaface.
by Yours Truly Jun 24, 2004
   
Urban Dictionary
Pizzaface
A person with alot of acne on their face
Wow!, Jessie has a pizzaface.
by John Apr 27, 2005
 
(Oxford English Dictionary)
pizza face, n.
slang and derogatory (orig. U.S. College slang). A person with facial acne.
1968 C. F. BAKER et al. College Undergraduate Slang Study (typescript) 173 Pizza-face, one with a bad complexion.
1970 J. BOUTON Ball Four 57 If we happen to see some fellow who is blessed with a bad complexion, we immediately call him something nice, like ‘pizza face’.
1992 Independent 27 Apr. 15/3 A teenager labelled by his peers as ‘old pizza face’ can carry the psychological scars around for life.
2003 N.Y. Times (National ed.) 13 Jan. C4/3 A less hostile, more comfortable communal gathering place for the geeks, outcasts and pizza faces.
   
11 August 1964, Delaware County Daily Times (Chester, PA), “Helen Help Us” by Helen Bottel, pg. 9, col. 2:     
“Pizza face”—pimply face.
(From a slang list submitted by “California Teen”—ed.)
 
29 December 1964, Oshkosh (WI) Daily Northwestern, pg. 17, col. 2 ad:
PIZZA FUNNY FACES. Pat out and trim Jeno’s Pizza into face shapes. Use onion slices, pimento olives, strips of mozzarella cheese for eyes, nose, mouth.
 
Google Books
The Choirboys
By Joseph Wambuagh
New York, NY: Delacorte Press
1975
Pg. 56:
“Blow it out your ass, pizza face!” Roscoe shouted to a sputtering acned man in a white Cadillac who was honking his horn and yelling as though he thought the…
 
Google Books
A Mile Above the Rim: a novel
By Charles Rosen
New York, NY: Arbor House
1976
Pg. 250:
“I’m gonna cut your eyes in a million pieces. I’m gonna make you a pizza face.”
   
Google Books
Firestarter
By Stephen King
New York, NY: Viking Press
1980
Pg. 88:
“Be careful we don’t decide to bother you next, pizza-face,” he said
 
Google Books
Christine
By Stephen King
New York, NY: Viking Press
1983
Pg. 209:
The pimples. The cries of Hey Pizza-Face!