Can chicken not completely cooked then cooled be fully cooked later?

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.

Internal temperature is one factor, but length of cook time is another. You just need to hit the right combination to kill all the bacteria.

There are lots of answers in this document: http://amath.colorado.edu/~baldwind/sous-vide.html

Assuming all the bacteria is dead, you still don't want to do this. Reheating it enough to finish cooking the center is going to make the outer edges super dry (i.e. chicken jerky).

Chicken must always obtain the temperature of 165F to properly kill salmonella. If you did not achieve that throw it out. If you got somewhere between 120 and 140 you killed many bacteria but not salmonella. The time it sat between 60 and 120 salmonella was multiplying at 140 it came to a halt, then from 120 back to 60 it started to multiply again. While the bacteria are multiplying they are releasing toxins. Cooking it to 165 or higher will kill the salmonella and the other bacteria but there will be to many toxins left behind. You can't get rid of toxins by cooking. In the ASSUMED time period there won't be enough time period to create enough toxin to kill you or even make you as sick as the original salmonella but it will make you more sick then you would like. Unless you are a professional chef with all the right equipment don't try "sous vide" they have bacteria test strips to ensure the proper procedures, cook times and temperatures. Yes you can cook chicken to only 131 for several hours but if the chicken wasn't prepared right and properly sealed food poisoning may occur. Get a good instant read thermometer and follow recipes properly. People who think they had the 24 hour flu actually received some sort of food or water poisoning in the last 12-28 hours, 6-48 for salmonella. Good luck on your next chicken and may it turn out perfect!!!

As long as the internal temp hits 165-170, you're fine. That should kill anything that might have grown in between. Chances are it's going to be a bit dry though.

I got food poisoning from doing that w/ a piece of ham. It was cooked thoroughly, and was at temperature, but I discovered if you partially cook something, and then, come back at a later time to fully cook it, that it needs to be super cooked or else you can get really sick!