People tend to bring this up as a gotcha against utilitarianism, with the premise that it's the same situation as the trolley problem rephrased and therefore should have the same answer, even though it "feels" more wrong, but I'm not convinced that these situations are actually equivalent and that the organ harvesting is actually the choice with the most utility. The reason being that, realistically, there are more factors involved that affect the utility of one's choices than just the six people who live or die.

For example: the operation is likely to be somewhat traumatic for the doctor, diminishing their quality of life, the people involved are not necessarily likely to live the same amount of time following the transplant because we haven't said anything about their age, general health (beyond that they need a transplant and are therefore likely to be rather ill), lifestyle (these factors can push utility in either direction as they apply to the donor as well), and the fact that transplants are not always successful. These people also all have friends and families whose lives will be impacted by what happens (this factor probably pushes for the transplant because more people probably have more people close to them). And finally and perhaps most consequently, if you set the precedent that you both have the ability and willingness to do this, you will likely create a fear in the general population that they or someone they know might be next on the donor list, because people have self preservation instincts and it's difficult to keep things secret forever. That kind of fear is a reduction in the well-being of all who learn about your operation, and though the amount is small, especially compared to a death, a small number multiplied across a significant chunk of the population that eventually learns about this results in a large net harm. It gets worse if anyone prone to paranoid thinking comes to the belief that they're "next" and attempts to sabotage their own health to make themselves a less appealing donor, or if people begun to mistrust doctors and avoid going to them when they would benefit from it- it's not difficult to imagine those last couple things ultimately causing some deaths, and it doesn't take more than a handful of those to counter your initial lives saved even without considering everything else I've just mentioned.

Now, something a bit similar can be said for the original trolley problem: those people are unlikely to have tied themselves to the track, so there's probably someone who has done it, and now people are going to be scared of some sadistic murder who goes around tying people to trolley tracks, as well as maybe distrustful of public transit infrastructure. However, in this case, this has occurred before the person at the lever arrived, and that damage has therefore already been done, and isn't relevant to their choice. With the organ harvesting scenario, it hasn't, and only occurs if the "kill the healthy person to save the sick ones" option is chosen.

This gets at a problem I have in general with these kinds of questions though: they usually either ignore the wider context of the scenario, which I believe is always going to be relevant to the answer, or they attempt to ensure it isn't relevant by modifying the scenario into one where no consequences exist except for the lives of the people involved, at which point the scenario can probably be contrived to make any ethical system result in counterintuitive choices with careful question design, but which also renders the hypothetical a bit pointless, became "your system leads to a result you probably don't like" doesn't mean very much anymore if the situation in which that happens is sufficiently unlikely to occur in real life.

A doctor can kill whoever they want. But only for science, of course.