My optimal breathing control pattern during the actual firing of a shot consists of preparing to take the shot at the bottom of my exhalation (breathing out). I breath calmly and consistently for a couple of breaths and then intend to take the shot at the bottom of my third breath - unless my breathing is not under control by the third breath - in which case the intended shot is coordinated with the 4th, or 5th, etc. breath in which I have gained control of my breathing. What I define as controlled breathing is when I am able to breath in, then out in a steady and consistent manner.
I had some long range tactical training with the US Army and without going into ad nauseum detail about connecting the mental and physical elements of breathing control, suffice it to say that taking the shot at the bottom of an exhaled breath - the point of normal pause - is when you're likely to have the greatest control because it is at that point that your heart beat, breathing, and muscle tension are all in sync and the least likely to cause an involuntary "jerk" during the trigger pull.
I also practice to be able to "take up the trigger" during the phase of letting my breath out so that at the point that the trigger is ready to break, my breath is at the bottom of exhalation where my normal breathing pause is at.
My suggestions for gaining control of breathing following exertion is twofold. First, practice good cardiovascular health by staying in shape for the type of exertion your likely to put forth. For the occasional hunter this means being able to hike around and up and down and not get so winded you have to stop and huff & puff. This can be especially challenging for folks who only get out to hunt one season or two each year.
Poor or minimal cardiovascular health puts you at a disadvantage even before stepping out of the truck. So I have to robustly agree with Rogue that this is very important.
My second suggestion is to practice, practice, practice. In the military we ran insane PT exercises while lugging our rifle on our backs, then we'd suddenly hit a spot where we had to drop, read the dope, and take the shot within a certain time limit (usually less than 30 seconds). This kind of drill forced us to practice getting our breathing under control as quickly as possible - while simultaneously acting as an intense cardio workout. I've been out of the military for almost 20 years now and I still run these kinds of practice routines to ensure that when I top out on a steep ridge - completely out of breath - I have the skill to drop into a stable shooting position and quickly get control of my breathing so I can take the shot.
Less than a week ago I ran this practice near dusk and took two shots at unknown distance. The second shot came after a particularly challenging climb through brush and it took me 5 breaths to pull the trigger. Both shots were first-round hits. That kind of practice gives me the confidence to take LR hunting shots under exertion that I can be confident are ethical. Besides having the skill to fill the freezer rather than watching the elk trot away, being confident that my shot is an ethical one is of utmost importance!