Adrenaline positive feedback.
The only way I see this happening is adrenaline having different functions in their bodies. Adrenaline is a hormone mostly responsible for the flight or fight response in the body, stimulating the heart and lungs to work more, constricting more external blood vessels to reduce bleeding, dilating the pupils etc. However, the stress caused by such response is usually associated with a slower healing process and reduced immunologic activity in humans, which is also why stress is bad for your health.
In your race however, unlike us who redirect the energy mostly for combat, higher levels of adrenaline has a positive feedback towards cellular division, stimulating mitosis and regeneration, which allows them to fight more, being more stressed and releasing more adrenaline. The harsher the battle, the more stressful the situation is, more adrenaline is released into the bloodstream and the more signaling to initiate mitosis the tissues receive. Regarding the numbing of the pain,adrenaline is already wonderful at doing that: enough adrenaline in your blood and you might not even notice a knife lodged in your back.
This process will come at a price though: 1-they will naturally have less energy available to fight, as a noticeable chunk of resources is being sent to damaged regions so that the cells can multiply at such speeds. 2-your creatures will need lots of measures to avoid mutation and cancer, because apparently it can regenerate lost structures to an extent and it can do it fast (regenerating fingertips within minutes is still a noticeable feat that will require cellular division at levels that might be beyond cancerous). 3-as their regenerative abilities are tied to positive adrenaline feedback, the more peaceful members of this species might have some problems healing wounds, so the healthier members might be the ones who are more stressed out and more easily aggravated ( might develop in a way similar to Spartans, as being stressed can be good for them). 4-they might be more susceptible to bleeding, as they can't afford as much to cut circulation to the tissues when they're in need of nutrients and energy to ensure successful regeneration, so they might bleed little more than an equally stressed human. 5-they will need a very efficient digestive system, as well as good ways to store the components necessary to regenerate lost structures.