The only organic control is grubbing, which requires that you get (practically) the whole root, not an easy endeavor.
Mowing always leaves a few inches above the ground, and that is generally enough for a mesquite to live indefinitely. Frequent mowing WILL, keep your place looking like it is free of brush , and can possibly be used to prevent effective seed pod formation, which would help to keep them from spreading.
As mentioned, Remedy + Diesel, definately works, but of course is not an organic method.
More organic, than above (lesser of an evil persay?) would be a cut stump method, cutting the tree to the ground, and then immediately applying diesel over the cut/exposed area, and around the base of the trunk.
Diesel is a petroleum product, but so is the stuff most all of us put in our vehicles.
Again, the most organic method is mechanical grubbing, but not something I recommend if you are on a shallow rocky soil.
Now to clarify a few things, no, keeping the few large mesquite around does not help (contrary to what many believe). I've got a 10 acre field we keep in mesquite, and where the young seedlings grow, is whereever they darn well please. If you like the looks of the large mesquite, you can surely keep them, but just know that the consequence of keepin any mesquite, is always having a source for mesquite beans, and therefore new mesquite seedlings.
Secondly, no, enhancing soil health will not kill a single mesquite. Mesquite are actually/typically found growing on better soil/ecological site.
To say that improving your soil health will improve your mesquite problem, would be like me saying that improving soil health will improve your tomato problem.
Weeds & noxious brush will grow in many place where we have a difficult time growing anything else, but that doesn't mean that they won't grow on really good soil too!
Best mgmt method to reducing new seedling mesquite in a pasture, is to keep as much grass cover as possible. And while improving soil can possibly improve grass cover, if the grass doesn't stay, then neither do the improvements. Ground cover is the key to reducing weeds and brush encroachment... Think of GRASS as your MULCH, in your flower bed.
Lastly, don't mow your mesquite, if you dont have to. Doing so, makes them harder to get rid of later on. They become multi-stemmed and almost immune to chemicals that would have worked otherwise, and when the next person or yourself stops mowing, then all you have is a really ugly, unmanageable field.