Reading through the first few years of the Los Angeles Free Press has already been very fruitful exercise! In addition to making (what I consider to be) a major breakthrough, I’ve started to find answers to some earlier questions too.
David Tudor (left) and John Cage performing at the 1971 Shiraz Art Festival. (source)
In my post from November 6, I tried to track down the source for a story that Zappa often told about John Cage. He recalled attending a concert in which he witnessed Cage drinking carrot juice with a contact microphone attached to his throat:
A short time after that, John Cage came to Claremont College and he was giving one of his … he does these performances with a throat microphone. He’d put this thing on his throat and drink a quart of carrot juice, or read something to you while he was drinking the carrot juice. 
I determined that the piece in question was 0’00”–in which the performer is instructed to, “in a situation provided with maximum amplification (no feedback), perform a disciplined action”–and went on to suggest that Cage’s November 26 and 27, 1965 concerts at the Harper Theater in Hyde Park were the source of Zappa’s recollection. Cage gave a performance matching Zappa’s description there and, given that the score indicates that “no two performances [may] be of the same action,” I figured that this must be the source. (Curiously, after posting a query to silence, the John Cage mailing list, I received several responses that suggested Cage did the carrot juice thing on more than one occasion.)
As it turns out, I was incorrect.
In the January 29, 1965 issue of the Los Angeles Free Press, Michael Agnello reports on a concert Cage gave with David Tudor at the Pasadena Art Museum, now the Norton Simon Museum (click on the image to the right to read the whole article):
It was a common swivel chair which the man (John Cage) sat on. Whenever he moved, I heard a grating sound of metal against metal. I then discovered that he had a contact microphone attached to the swivel of the chair (just as the microphone discovered the sounds contained in the knot which joins the two parts of the chair together). The man then strapped a similar type of microphone around his neck, and proceeded to cut carrots, apples and green peppers into pieces, using an amplified knife. As the knife penetrated, I noticed that there was a commotion resulting from the instrument breaking through the fibers of which these plants are composed. My eye saw and my ears heard and at one point I was moved to say to myself, â€œThis is ridiculous, what possible purpose could there be to all this madness?â€ At that point a fellow named Snyder began to expound a variety of attempted witticisms; no, he was not intended to be part of the show—–and my eyes saw and my ears heard. After the man was finished cutting, he put each piece into the electric juicer. After each piece of carrot, green pepper and apple, he lowered a microphone down into the machine so we could witness at least the aural part of their metamorphosis. Altogether, it was a happy experience and it fittingly concluded with the sound of the juice passing through the manâ€™s throat. 
The description matches Zappa’s perfectly. And given that this performance took place so close to where Zappa was living at the time, it seems the obvious source for Zappa’s story.
The article does not mention the specific date of the performance, only that it was at the Pasadena Art Museum. A quick search on ProQuest revealed the following confirmation:
John Cage, one of the country’s new musical experimenters, will present three of his compositions at the Pasadena Art Museum in two performances, 7:30 and 9:30 p.m., Thursday, Jan 7.
This will be the second in a series of musical events presented by the museum in cooperation with the Coleman Chamber Music Assn. and the Pasadena Music Teachers Assn.
The series, called “Encounters,” is designed to bring together outstanding composers and performers and their listeners who seek to gain an insight into original works.
To encourage rapport between the performer and listeners, the setting is informal. The compser or performer will speak before his works will be played.
Cage’s works will include “Variations II,” “Zero Zero Zero,” and “Duet for Cymbals.” Pianist will be David Tudor. 
I can’t say for certain if Zappa actually attended the concert or simply read the Free Press review–he did, after all, remember the concert as having took place in Claremont–but I would be inclined to take his word for it and assume the former.
FURTHER UPDATE (December 11, 2009):
In trying to track down some more information about Cage’s performance, I took it upon myself to contact the Pasadena Art Museum (now the Norton Simon Museum). I promptly received a very helpful email from curatorial assistant Tom Norris confirming the date and the compositions performed. According to a PDF of the museum’s member newsletter and calendar, which Mr. Norris was kind enough to send me, Cage gave:
Two performances January 7, Thursday evening: 7:30 p.m. and 9:30 p.m. An encounters concert featuring the composer and David Tudor, pianist. Tickets $2.50, students $1.50.
Not a bad deal! The newsletter describes the program thusly:
Whenever contemporary experimental music is discussed, the name of John Cage is sure to be mentioned. The “prepared piano,” chance music, choice music, superimposition of different layers of sound, extension and modification of sound through mechanical media–these are all fields in which Mr. Cage has taken the lead. Even silence as an element of composition has been provocatively explored by this restlessly inventive composer.
Encounters will bring John Cage and David Tudor to the Pasadena Art Museum on Thursday, January 7th, for two performances at 7:30 and 9:30 p.m. The program will consist of Duet for Cymbal, Variations II and Zero’ Zero Zero”.