2004 Santa Fe stalling problem

A few weeks ago, when I stopped my 2004 Santa Fe at the end of my driveway, it stalled and I couldn't get it started. I called a wrecker and had it towed to a local garage. They told me that the air filter was packed with sunflower seeds by a rodent. Seeds were removed, car was checked for further problems, and none were found. Problem solved, or so I thought.

Yesterday, I was driving my car when the engine quit. I had just left home and driven approximately 1 mile. Luckily, I was able to coast into a rest area. As before, I was unable to restart it at first, but after letting it sit for approximately 1/2 hour, I tried again and this time it started successfully. I drove it home without incident where it sat for the remainder of the day. This morning I drove it to work (about 10 miles) without incident as well.

What is my problem?