Can chicken not completely cooked then cooled be fully cooked later?
I cooked some chicken but didn't realise it hadn't completely cooked through. It then went in the fridge and a day later I cut through it to discover the problem.
Is it now safe to put that chicken in the oven to finish cooking it or am I risking major health issues?
Please don't bother.
Meat spending any significant amount of time in the danger zone of 40 F - 140 F (4.4 C - 60 C) that has not been fully cooked should never be re-chilled to be served later. Given that Salmonella can be found throughout poultry, not just on the surface, and that it's almost guaranteed that your poultry had some level of a pathogen present, you shouldn't risk it.
Your not so thorough cooking, likely to an internal temperature of 120 F - 130 F, has basically heat-shocked the bacteria present imbuing them with a much higher than usual heat resistance. On top of that it likely took at least a few hours to get the meat back below 40 F (at which they can still grow, albeit slowly), so depending on the initial amount of infection and the time spent in the danger zone you likely have a bacteria population minimally in the hundreds of millions, possibly in the billions.
In this answer of mine I give a lot of detail regarding the relation between temperature and duration and it's effect on Salmonella. The important take-away is that with any reasonable temperature death is not instant, it's merely a percentage of the population that is killed at any given temperature and duration. As few as 100,000 Salmonella cells can get you sick, and it takes much fewer for E. Coli. So even if you're killing five-9's worth of bacteria, is it really worth it?
I'm all for eating rare meat (not poultry), but only if I'm confident in the quality and handling of said meat before it reaches my plate. Think of it this way, if the meat were exposed to these conditions before it reached your plate it would be considered gross negligence, and would likely result in a recall.
The guidelines that many agencies publish to safely cook meat all assume typical levels of contamination, given proper handling (though they do err greatly on the side of safety). They simply aren't accurate when you are starting with meat that has a population large enough to sicken or kill a small village.
I'm not even going to begin to address the toxic waste products produced by some pathogens, which are not destroyed by heat.
Throw it out and prevent this in the future by being sure to cook it all the way through. It sounds like you likely just grabbed the chicken out of the refrigerator and threw it directly on the heat, this can lead to the exterior cooking too quickly before the interior has time to cook. Get in the habit of setting your meats out for 30m to an hour so that it reaches room temperature throughout, but cook it immediately, do not re-chill it.