Now that rain has returned, your lawn should be greening up. It may not be perfectly healthy yet, but there is probably more green than brown. So, give it as much time as you can to recover before you absolutely have to mow.
If your lawn is long and green, go ahead and mow. I prefer using a walk behind mower to a riding mower right now. Granted, it will be a bit more tiring, but it will be better for your recovering turf.
Taking some time to rake out any brown, dead grass is also a good idea. It will give new plants space to grow. However, keep an eye on the brown or bare spots. If they remain small, the surrounding grass should fill them on its own. If they are large, however, you will have to overseed.
Don’t overseed now while we still have temperatures in the 80s and above. It is possible to have another dormant period this summer if the rain doesn’t continue, in which case, you will have to prepare the soil and overseed again. While you wait for temperatures to come down and rainfall to become more predictable, watch those bare spots to be sure no weeds take over. Keep these spots clear of weeds by digging them up.
When weather conditions become more conducive to growing turfgrass, rough up the soil with a metal rake, plant grass seed, and then scratch it into the ground with your metal rake, followed by a good watering. Be sure these reseeded spots receive at least an inch of water a week either from Mother Nature or from a sprinkler. The new plants should be well established when you apply your fall fertilizer, or have your lawn service apply it.
If you have area in which the grass is green but thin, these areas should also be overseeded. By early autumn, you should have a nice, thick carpet for your whole lawn.