There is a (true) theory of corporate management that (accurately) describes why CEOs of big companies are paid so much, and also why they get hired and fired seemingly as frequently as losing NFL coaches. Horace Dediu describes it like this: "They demand and are unquestionably given absurd pay that has no relationship to performance. Such pay has no relationship to performance because it isn't designed to reward performance but to account for the risk of arbitrary and very public sacrifice. Boards (and hence shareholders) are deliberately hiring a scapegoat for sins as yet unknown."
In this formulation, CEOs are paid 331 times what their average workers are not because of their magical management skills, but because they can one day be sacrificed to the gods of the free market, thereby cleansing the corporation itself and the rest of its managers from blame for when things inevitable go wrong. Some CEOs are good and some CEOs are bad, but shit happens regardless, and when it does, they will be fired, so they gotta get that $$$ as fast as they can. Assured eventual public humiliation demands rich compensation.
Don Thompson lasted less than three years at the head of McDonald's. During that time, the company's stock price went down, sales declined sharply, and the company was hit with legal rulings ordering it to clean up its labor practices—which, by the way, became the target of a huge national protest campaign, during the tenure of Don Thompson. There is little evidence that Don Thompson, who was paid almost $10 million in 2013, was a "good" CEO.
That is beside the point. The real question is not whether Don Thompson was worth $10 million a year. The real question is whether anyone can be a "good" CEO of a company like McDonald's, without changing its food quality, its worker pay policies, its menu, its ubiquity, its encouragement of sloth in the populace, its sickening branding—everything about it, essentially. Can you be a good CEO of a bad company? It seems doubtful. But we won't learn the answer from Don Thompson, a bad CEO.