“Un oeuf” (French for “one egg”) sounds like “enough,” prompting the pun:
 
Q: Why does a Frenchman have only one egg in his omelet?
A: Because one egg is un oeuf.
 
The joke dates to at least the 1870s. “Why should a French boy never eat more than one egg ? — Because one egg is un oeuf” was cited in 1872. “How many eggs can a Frenchman eat? One, because one egg is un oeuf (enough)” was cited in 1879,
 
The specific mention of an omelet is more recent. “Do you know why the French make an omelette with only one egg? Because one egg is un oeuf” was cited in 1982.
 
   
Google Books
Little Folks
Volume 4
1872
Pg. 31:
Why should a French boy never eat more than one egg ? — Because one egg is un oeuf (enough).
 
Google Books
October 1879, The Poultry World, “Pleasantries,” pg. 313, col. 1:
How many eggs can a Frenchman eat? One, because one egg is un oeuf (enough).
 
Google Books
10 April 1902, The Independent, “Pebbles,” pg. 880, col. 2:
“Why do the French eat only one egg for breakfast?” “Can’t guess. Why?” ” Because one egg is un oeuf.” — Sphinx.
   
Google Books
The Atlantic Monthly
Volume 166
1940
Pg. 507:
The thought of food recalls the conundrum as to why a Frenchman rarely eats more than one egg. Because, having eaten that, he has had un oeuf.
 
Google Books
The Comic Encyclopedia:
A Library of the Literature and History of Humor Containing Thousands of Gags, Sayings, and Stories
By Evan Esar
Garden City, NY: Doubleday
1978
Pg. 82:
A Frenchman seldom eats more than one egg because, having eaten one, he has had un oeuf.
 
Google News Archive
18 January 1982, Eugene (OR) Register-Guard, “Words” by Michael Gartner, pg. 5B, col. 2:
“Do you know why the French make an omelette with only one egg?
“Because one egg is un oeuf.”
 
Google Groups: alt.games.sf2
Super Duper SF2ce…
Noah Hanna
10/23/92
(...)
“In France you only need one egg because one egg is un ouef”
-My french teacher
 
Google Groups: bit.listserv.techwr-l
Bilingual Puns
Lester Klein
8/24/94
Why does a Frenchman have only one egg in his omelet?
Because one egg is “un oeuf”.  It’s a bilingual pun because “un oeuf”, French for “one egg”, sounds like the English word “enough”.
 
Google Groups: alt.humor.puns
Egg puns
Steve Poggio
11/6/94
Subject: Egg puns
> Is it true the French generally have a small breakfast
> because they find that one egg is un oeuf?
 
> That’s the worst yoke I’ve heard in a long time!
 
Which is eggsactly why you shouldn’t make egg puns!
   
Google Groups: rec.food.cooking
Food and Cooking Humor!
Dan Masi
8/22/96
Charles Demas wrote:
> One of the first bilingual puns was:
>
> Why can a Frenchman make an omelet with just one egg?
>
> Because one egg is enough!
>
> For those of you who don’t get it,  enough = a neuf(egg in French).
 
For those of you who *still* don’t get it, try
“Because one egg is un oeuf”.
 
(“un oeuf” == “an egg”.  “a neuf” ==  “at nine”, which I suppose would also work…  😊  )
   
Google Groups: alt.humor.puns
Puns of the Weak 03-15-04
Stan Kegel
3/16/04
(...)
French waiter:  “What would you like for breakfast?” Customer:  “One egg is un oeuf.” (Jim Ertner)
 
Google Books
Jokes and the Linguistic Mind
By Debra Aarons
New York, NY: Routledge
2012
Pg. 179:
Q: Why do French people only have one egg for breakfast?
A: Because one egg’s un oeuf.
 
This is a rather obvious made-up joke, the purpose of which is to demonstrate the point that the sequence un oeuf, which means “one egg” in French can be reanalyzed as the English word, enough.
 
YouTube
One Egg Is Un Oeuf
lerpinez
Uploaded on Jan 19, 2012
One egg is un oeuf.
(From the television drama West Wing, May 10, 2000.—ed.)