The humorous British- and Irish-English phrase to extract the Michael (or the michael) from (or out of) is a polite variant of the informal phrase to take the mickey out of, meaning to tease or ridicule.

Similarly, the humorous Irish- and British-English phrase to extract the urine from (or out of) is a polite variant of the slang phrase to take the piss out of, meaning to tease or ridicule.

The phrase to extract the Michael (or the michael) from (or out of) occurs, for example, in Irreverent slant on Irish life, by Hedley McConnell, published in the Bray People (Bray, County Wicklow, Ireland) of Thursday 1^st April 1999—RTÉ (Raidió Teilifís Éireann) is Ireland’s national radio and television broadcaster:

‘Don’t Feed The Gondolas’ is, in this writer’s opinion, one of the most entertaining home-grown programme [sic] to have appeared on RTE in over 37 years of broadcasting.
According to information gleaned from their website, the rather peculiar name evolved from a meeting of an Irish County Council, at which it was proposed that gondolas be placed on some nearby lakes as a tourist attraction.
Apparently one of the councillors became concerned about the suggestion. ‘Gondolas are all very fine and well but who is going to feed them?’ he asked. This, in effect, sets the tone of the show.
In this age of so-called ‘political correctness’, the Irish are still the butt of many a ‘Paddy’ joke. (One wonders would the problem—if, indeed, it is a problem—still exist if we were all black?)
A more politically incorrect trio than Sean Moncrieff, Brendan O’Connor and Dara O’Briain it is difficult to imagine. The three gondoliers are dedicated to extracting the Michael from most, if not all things Irish—including people—and while this may not be to everybody’s taste, at least it is Irish people who are doing it, which makes a nice change.

The earliest occurrences that I have found of the phrase to extract the Michael, also to extract the michael, are as follows, in chronological order:

1-: From the account of a match that was played at Sittingbourne between Sittingbourne Football Club and Sheppey United Football Club, published in the East Kent Gazette (Sittingbourne, Kent, England) of Friday 3^rd February 1956—Sheppey is an island off the coast of Kent:

With the sudden departure to Gillingham of Dave Hawkins, the Islanders’ attack was led by our old friend Bert Lightning. With very little help he never looked like worrying McGuinness and switched to the right wing after the interval, when home supporters amused themselves by good naturedly extracting the Michael from the former Sittingbourne favourite every time the ball went anywhere near him.

2-: From an advertisement for Edwards Television, by Roy Edwards, published in The Reading Standard (Reading, Berkshire, England) of Friday 23^rd March 1956—here, the noun geyser denotes a domestic water-heater:

We want to clear some of the second-hand sets that are taking up space around the place, some of them are very good, some of them a little dodgy, some of them would be suitable for presentation to the nasty little boy next door, in the hope that he might either electrocute himself or blow himself up. […]
Just as the butcher has dozens of people each day who say “Is it tender?” we have our customers in this trade who have their stock remarks. Most of them occur in connection with service, of course. There is always the customer who says “There can’t be much wrong with it, we’ve had it for four years with no trouble at all.” Quite frequently, the customer is right, but the basic assumption tickles us. Here are a few more stock remarks. “Will you please test these 17 valves and then tell me where to put them all back?” “My nephew, who knows a bit about electricity, dusted the inside of the set last night, and now we only get half a picture.” “While you’re here, I wonder if you could just look at my hair-dryer—geyser—fuses—refrigerator—piano—canary.”
If any of you dear readers are blushing a little, please don’t think I’m extracting the Michael too much: I can assure you—it’s all in fun!

3-: From Oxford Revue Better Than Ever Before, published in the Edinburgh Evening News (Edinburgh, Midlothian, Scotland) of Friday 24^th August 1956:

Edinburgh Town Council is only one of many victims pricked by a satirical pin in “Better Late . .” a revue staged by Oxford Theatre Group at their new base in Gullan’s Close. […]
Nothing escapes in the great game of Extracting the Michael. Jazz, Jack Spot, the Festival itself, Big Brother, “B and K,” sex, and “Teddy” boys are just some of the victims.

4-: From the column Cables From New York, by Rex North, published in the Sunday Pictorial (London, England) of Sunday 26^th May 1957:

THE MICKEY

HEY, there, bud. Mickey Cohen, self-confessed killer, is a Mickey taker.
And he has been extracting the Michael from that evangelist Billy Graham.

5-: From the column Off the record, published in the Worthing Gazette (Worthing, Sussex, England) of Wednesday 27^th November 1957—the Goon Show was a British radio comedy programme; the U.S. musician Spike Jones specialised in spoof songs:

We’ve got the Goons, but America still has Spike Jones, and H.M.V. presents an EP by the King of Corn called “Fun In Hi-Fi.” Aimed, generally, at extracting the michael out of modern gadgets, the record includes a glimpse of scatty science fiction as well.

6-: From You, too, can be a Quiz kid!, by Rushworth Fogg, published in The People (London, England) of Sunday 2^nd February 1958—the author was a contestant in Criss Cross Quiz, broadcast on the ITV (Independent Television) network:

Sitting in my armchair I would have known most of the answers to questions I fluffed. It’s much harder standing up with a hot light overhead and a quizmaster—even a nice chap like Jeremy Hawk, who never extracts the Michael—waiting expectantly.