I think it depends on what you are going to plant.
For
root vegetables, I would want it very well rotted, even when placed a foot down. I would keep turning the pile until it starts to cool down and look more like soil. I believe that means the excess nitrogen has been used up by decomposing bacteria.
For above-ground vegetables that are nitrogen lovers, I think the compost could be pretty "hot." Depending on the amount, I wonder if you need to compost it at all -- provided you add the carbon/brown matter you mentioned.
My 2 cents. Others here will chime in with more authority.