One thing you might consider is did you actually run out of gas, or was it low and you THOUGHT you ran out of gas? If you are sure you ran out of gas, sometimes it takes a few minutes for the fuel to fill the bowl. If I run my edger out of gas intentionally, then it won't start for 2 or 3 minutes after I put fresh gas in and open the valve.
As a rule of thumb, before pulling anything apart, figure out what is missing - fuel, spark, or compression. Fuel you can squirt a teaspoon into the carb with the air cleaner off. Spark you can use a spark tester or pull the plug, for compression you can hear it if electric start or feel it if it is rope pull. You can also put your thumb on (not in) the spark plug hole. If it is flooded, pull the spark plug and pull it thru a couple times then put the plug back in. Once you know why it doesn't run then take things apart. If you dive right in the original problem may end up compounded. Also, a good friend had an engine that would not start. His dad handed him a gallon of clear kerosene instead of gasoline. He didn't figure it out for about 4 hours of could it be this, could it be that?