Originally and chiefly American English, EGOT is an acronym from the initial letters of Emmy, Grammy, Oscar and Tony.
– Emmy: any of the statuettes awarded annually to an outstanding television programme or performer by the American Academy of Television Arts and Sciences
– Grammy: any of the statuettes awarded annually for outstanding achievement in the recording industry by the American National Academy of Recording Arts and Sciences
– Oscar: any of the statuettes awarded annually for outstanding artistic and technical achievement by the Academy of Motion Picture Arts and Sciences
– Tony: any of the medallions awarded annually by the American Theatre Wing, New York, for outstanding achievement in the theatre

The acronym EGOT denotes the Emmy, Grammy, Oscar and Tony awards viewed as a single achievement.

It was coined by the American actor and musician Philip Michael Thomas (born 1949), whose most famous role is that of detective Ricardo Tubbs in the television crime drama series Miami Vice (1984-89).

EGOT is first recorded in an interview of Philip Michael Thomas by Associated Press, published in newspapers nationwide in August, September and October 1984—the earliest instance that I have found is from the San Francisco Chronicle (San Francisco, California) of Saturday 18^th August 1984:

On his 35^th birthday last May, he said he launched a plan called “EGOT.” “That stands for Emmy, Grammy, Oscar and Tony,” he said. “Hopefully in the next five years I will win all those awards.”

The second-earliest occurrence of EGOT that I have found is from Miami: Nice!, an interview by Tom Jicha of Philip Michael Thomas and Miami Vice co-star, Donald Wayne Johnson (born 1949), published in The Miami News (Miami, Florida) of Tuesday 6^th November 1984:

Thomas appeared in the original Broadway production of “Hair” and later played a lead role in the San Francisco version for 18 months. A musical he wrote, “The Legend of Stacker Lee,” has been produced in Los Angeles and he is president of his own music company, PMT International.
The company title sounds somewhat pretentious, but it pales compared to the show business goals Thomas has set. He has a codeword for them: EGOT. It stands for Emmy, Grammy, Oscar and Tony. “I’ll get the first one here [for “Miami Vice”] and maybe the second, too, [for the album]. That might seem cocky, but I believe in me.”
[…]
He is working on converting the back room of his trailer into a music studio. He has completed 10 songs for a middle-of-the-road album he hopes to release shortly.

The acronym has come to also denote a person who has won the Emmy, Grammy, Oscar and Tony awards. This is first recorded in the magazine Jet (Johnson Publishing Co. Inc. – Chicago, Illinois) of Monday 20^th May 1985:

TV star Philip Michael Thomas, of “Miami Vice,” who celebrates his 37^th  birthday on May 26, says his goal this year is to become an EGOT—winner of an Emmy, Grammy, Oscar and Tony awards. He says his upcoming album, Living The Book Of My Life, which he wrote and produced, will cop the Grammy. He will then pursue goals that lead to the Oscar and Tony.

photograph of Philip Michael Thomas on the cover of Jet – 20^th May 1985:

 

Philip Michael Thomas’s egotism soon became an object of ridicule. The following, for example, is from Stargazing: A potpourri of patricians, plebeians and peccadilloes, published in The Kansas City Star (Kansas City, Missouri) of Friday 6^th December 1985:

When, inevitably, Quotations From Philip Michael Thomas, is published, it will contain the inimitable wisdom of a TV star who has been called “the world’s most unrestrained mouth.” Among the more memorable gems:
● “The most incredible person I’ve ever met is myself. There’s a never-ending well of expression that is coming forth from me. My art is forever.”
● “I am the new blood.”
● “I’m a pure artist. I represent art in life. There are no limitations based upon this creative essence that flows from me.”
● “I am Christmas.”
Well, what can you expect from a TV star whose new album includes such humble ballads as “I Am So Beautiful”?
What can you expect from a TV star who wears a medal engraved with the letters EGOT, which stand for “energy, growth, opportunity and talent” or for “Emmy, Grammy, Oscar and Tony,” depending on when you ask?

Philip Michael Thomas’s coinage of EGOT and egotism naturally led to the association of the acronym with the noun ego, denoting a person’s sense of self-esteem or self-importance.

The following, for example, is from the review of Miami Vice, the 2006 film based on the television series, published in the column Ebert at the Movies, in the Jackson Hole News and Guide (Jackson, Wyoming) of Wednesday 2^nd August 2006:

Thomas’ ego was such that he sported a necklace that spelled out “EGOT” – for Emmy, Grammy, Oscar and Tony. Last I checked, he’s only two consonants and two vowels shy of reaching his goal.