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.
Mushrooms should be stored in the refrigerator in a breathable container. A paper bag works great. Mushrooms stored in this manner should last at least 4-5 days.
For the record, the top of the fridge is one of the worst places to store foods. Depending on the model it can actually be a few degrees higher than ambient temperature, thus accelerating spoilage.
Best Answer
Shiitake mushrooms are certainly not poisonous, either raw, cooked or as you posit in your question, after being cooked, stored in the refrigerator and consumed later.
If you enjoy eating them cold, go for it. Just keep in mind that like any other cooked food, they can spoil if not stored properly (I have some leftover Chinese mustard greens stir-fried with shiitakes in my fridge right now, and have been eating them for days).
However:
There is a rare allergic reaction in some people called Shiitake Dermatitis which is brought on by eating uncooked or under-cooked Shiitakes. Note that this does not mean it is poisonous, it is a food allergy for all purposes like any other (think shellfish, wheat, soy etc.)
In addition to that,
If you look, you will probably find advice to cook all mushrooms before eating (this probably has more to do with digestibility than anything though).
Although I believe this is really aimed more at wild harvested mushrooms, in his wild mushroom identification book "All that the rain promises and more", David Arora can be quoted as saying:
Personally, I would not shy away from eating Shiitakes raw for any safety reason, but you then may be robbing yourself of some of the nutritional value.