I haven't been able to find this record by searching. Obviously England's first innings in the current Ashes test is unusual, with Cook scoring 244, and the other 10 Englishmen scoring only 235 (12 extras prevent Cook from getting most of the runs in the innings). But what is the highest test innings in which one batsman has scored more than all the others combined?
For the sake of this answer I'm going to assume that "Highest test innings" is referring to the highest team score in which one batsman scored the majority of the runs. As it turns out, it doesn't change the answer much - pretty much the only thing that changes is the year!
If you don't care about all eleven batsmen getting out on the ground at some point, there's a pleasingly elegant answer: It's Brian Lara's 400* against England in 2004, part of a West Indies innings that declared at 5/751. Jayawardene came close to beating that record in 2006 when he top scored with 374 runs in an innings of 5/756, but he scored one boundary too few for his knock to comprise 50% of the innings.
For a completed innings, the record holder is the same as the record holder for every other batting-related record: Sir Don Bradman, who was responsible for 304 of Australia's first innings total of 584 in the 4th Test of the 1934 Ashes. Remarkably, four years earlier in Bradman's inning of 334 against England, he scored 59% of a total team score of 566 (including 14 extras), which is the highest score by an individual that comprised more than 50% of a completed innings total.