Like most other Wham! songs from this period, it was written by George Michael. He discussed in the fall 1985 issue of ASCAP in Action: “It’s the only song I’ve written that successfully came from a backing track first. I wrote the Linn drum pattern and found a synthesizer program I liked and wrote the backing track in one evening, took it back to the hotel, and wrote the vocal in a hotel room the next morning. Because it was thrown together that way, I never looked at it as a single 'til everybody started saying it was great.”[4] …

“Everything She Wants” was recorded in August 1984 at Marcadet Studios in Paris using a 3M 32-track recorder and was most likely finished off at Sarm West’s Studio 2 in London after most of Make It Big had been finished at Studio Miraval in southeastern France over a period of six weeks.[2][5] The song was based around a mono 2-bar LinnDrum loop, originally intended as a guide but it was eventually kept on the finished record.[6] Despite being credited as a Wham! track, it’s also a departure in the sense that Michael recorded the track entirely by himself (with engineer Chris Porter in attendance), as opposed to the typical process of involving session musicians and Michael conveying his ideas to them. Apparently, the musicians involved in Make It Big had already flown back to the UK when Michael came around to recording the track en route from Studio Miraval, so it was partly out of necessity he worked on it himself. According to Porter, “I think this was when George started to realise that if he wanted to, he could do everything himself. He could cut out all these other people and their ideas.” This would pave way for “Last Christmas”, which followed the same approach.[7]

https://en.wikipedia.org/wiki/Everything_She_Wants