It does say that if fat turned into jelly, and you warm up the chicken to the point where the jelly fat turns back into liquid, it's a problem. Also, Tag's soap example doesn't work, because no chemical change occurs in the soap, the outer layer is simply washed away. I never thought about this issue, but maybe there would be a problem, consult a Rav.
My water example would say it's not allowed, but then again, you're not allowed to take an egg and cook it on the hood of your car, so maybe your hand can be considered using an item to cook, so it's not very clear.