Ask Mr. Smarty Plants

The US Forest Service describes the effect of fire on Prosopis glandulosa (honey mesquite).  Range fires can reduce the mesquite canopy but usually does not kill them.  Around 90% of the trees survive to resprout from the roots or crowns. The report also shows that using a propane burner at high temperature (>780 degrees F) was successful in killing 100% of young trees (<1.5 years old).  Burning after treating with herbicide (2,4,5-T) resulted in mortality in 24-32% of the trees that had been top-killed by the application as long as four years before.  The explanation for the high percentage of mortality was the fuel provided by the dead material on the treated trees that increased the fire intensity.

The Mesquite Savanna Project at the Vernon Texas A&M station uses herbicides and prescribed burns to manage the mesquite growth form so that there is less foliage on the trees giving grass a better chance of growing beneath the trees.

You might also be interested in reading How to Beat Mesquite from Texas A&M Brush Busters.

Finally, I consulted the Wildflower Center's ecologist, Dr. Mark Simmons, and he said that he hadn't had any experience using propane torches on mesquite. He suspects, however, that there would be meristematic tissue safe below the surface that would allow resprouting. He uses a propane torch for weed control and many perennial herbaceous weeds (including grasses) can resprout after treatment. He thought it certainly would be worth trying but he isn't optimistic that is going to kill all your mesquite.