About half of our St. Augustine lawn appears dead after this brutal summer. The section that survived was established earlier and gets less sun. Both areas received the same amount of water. Is it a good idea to spread compost over the brown area this fall? If so, when? Any other ideas would be welcome.

Compost is a good way to revive a lawn but only put 1/4 to 1/2 an inch on.

No more than 1/4 inch of compost. That amounts to 1 cubic yard per 1,000 square feet. If you use more you can easily smother the turf.

Not sure that will do anything for your grass, though. Dead St Aug cannot be revived. You need to change the water your lawn gets watered. The dead part needed to be watered more deeply (longer). I am assuming you were watering no more than once a week in the summer. If you were watering more often, then that could be your problem. Deep roots come from deep watering. Shallow roots come from frequent watering and are much more susceptible to dry weather.