“Capon” sounds like “cape on.” “It’s so cold out there today, I saw a chicken with a cape on” is a joke that was printed in a 1976 newspaper. “It was so cold I saw a chicken with a cape-on!” American entertainer Glen Campbell (1936-2017) told an audience in 1984.
 
           
Wikipedia: Capon 
A capon (from Latin caponem) is a cockerel or rooster that has been castrated to improve the quality of its flesh for food and, in some countries like Spain, fattened by forced feeding.
 
In the United Kingdom, birds sold as capons are chemically or physically castrated cocks.
       
5 February 1976, The Morning News (Wilmington, DE), “Transplanted New Yorker defers to Delaware weather” by Betty Burroughs, pg. 39, col. 3:
“It’s so cold out there today, I saw a chicken with a cape on.”
(Said by some wag at a bar.—ed.)
   
25 March 1984, The Tennessean (Nashville, TN), “Glen Campbell Rekindles Country Music Relationships” by Robert K. Oermann, Showcase, pg. 7, col. 3:
“It was so cold I saw a chicken with a cape-on! (Capon, get it?)
(Said by Glen Campbell.—ed.)
 
15 March 1986, The Desert Sun (Palm Springs, CA)< "National Hockey League notes" by Ken Rappoport (AP Hockey Writer), pg. H11, col. 1:

One of the jokes told at a recent Rangers team photo session:
 
“It was so cold last Friday, I saw a chicken with a capon.”
 
17 December 2001, Buffalo (NY) News, “Reporters’ Notebook,” pg. B2:
“It’s so cold out, I saw a chicken with a cape on.”
   
Google Books
Yes You Can!:
Behind the Hype and Hustle of the Motivation Biz
By Jonathan Black
New York, NY: Bloomsbury USA
2006
Pg. 197:
“It was so cold last night…,” she pauses meaningfully, “... that a chicken crossed the road with a cape on.”
 
Dead silence.
 
“Capon, get it?” A few hearty laughs. “I got that from Garrison Keillor,” she confesses.
   
Twitter
FreakNasty Ferretti‏
@FerrettiThawts
Last night it was so cold, I saw a turkey with a cape on!
#getit
4:53 PM - 23 Jan 2012
 
Twitter
Kiefo “Maths” Nilsson‏
@konilsson
Replying to @NekoCase
@NekoCase Glen has a joke he says when it is cold it goes like this “It’s so cold I saw a chicken with a cape on”.
2:05 AM - 25 Jun 2012
     
Reddit—Delaware
It was so cold today in Sussex County, I saw a hen with a cape on. self.Delaware
Submitted January 7, 2014 by burntswamp🍰
That’s cold!
 
Twitter
Bob Mahar‏
@BobMahar
Boy, its cold outside!
“How cold is it?”
Its so cold I saw a chicken with a cape on.
11:47 AM - 9 Nov 2014
   
Twitter
VicedRhino‏
@VicedRhino
Replying to @PapaBirdJake @biblereloaded
It was so cold out today, I saw a chicken with a CAPE ON!!! Capon, get it? get it?  IT’S FUNNY DAMMIT!! 😭😭😭
8:39 PM - 30 Mar 2017
 
Boston (MA) Globe
It’s so cold that . . . we want your jokes
By Martin Finucane GLOBE STAFF JANUARY 02, 2018
(...)
It’s so cold out I saw a chicken with a cape on. — Several readers (not clear why everyone loves this classic)