Many times I have heard someone having to pull the plug because the patient was declared brain dead before the body healed fully? Why do they pull the plug without first waiting till the body/brain has a full recovery? What situations if not recoverable would they allow the body to fully heal before pulling the plug?
Brain death is permanent damage to the brain. Depending on the specific definition, it may refer to either cerebral death or to failure of the brainstem to maintain functions like respiration.
It could be possible for mistakes to be made and for brain death to be declared in a patient who has not actually experienced brain death. However, this is incredibly incredibly rare: zero cases have been reported when appropriate guidelines are followed (Wijdicks et al. 2010). If someone is diagnosed as brain dead after a trauma, the medical staff involved in making that decision are stating that there is effectively no chance for recovery. Brain death is not diagnosed lightly.
Wijdicks, E. F., Varelas, P. N., Gronseth, G. S., & Greer, D. M. (2010). Evidence-based guideline update: determining brain death in adults: report of the Quality Standards Subcommittee of the American Academy of Neurology. Neurology, 74(23), 1911-1918.
@Bryan Krause answered your question prior to the edit. Your new question is relatively easy to answer.
Why do they pull the plug without first waiting till the body/brain has a full recovery?
Some organs have a high regenerative capacity (you can donate part of your liver and it will mostly grow back), some have a high compensatory ability (you could live with one lung), etc. However, the brain isn't one of those organs.
Until relatively recently, it was thought that the brain had no regenerative ability at all. We now know that to be untrue. Still, compared with most other organs, the brain doesn't "heal" well.
It's one thing to keep a comatose patient alive (e.g. after a bad stroke) to see if time will help. Neurons do form new synapses that allow for a degree of recovery. But most times with a stroke, only part of the brain is involved, and the brain does have compensatory abilities (new synapse formation.)
But brain death is much more than a coma.
When there is so much damage that a patient can be declared to have undergone brain death, people don't recover from that. There isn't any reason to let someone heal all their other organs - or the brain - fully when the one organ that makes us who we are is dead and the only way to stay 'alive' is by artificial means (ventilators, total parenteral nutrition or a g-tube, etc.)
What situations if not recoverable would they allow the body to fully heal before pulling the plug?
Family inability to allow a hospital to "pull the plug" leads to prolonged life support and recovery time for such patients. This is not necessarily good, however. Some patients who were declared brain dead (less rigorous criteria) survived post life support, but none has gone on to become functional (and usually stay in a vegetative - or minimally conscious - state at best.)