Small Engine Troubleshooting Problems? Ask a Mechanic Now.
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This sounds like you have a heat issue. Once you shut the engine down it will build more heat in the engine area due to no air flow, this could be heat soaking the ignition and causing it to quit firing even with a new coil. Once it has a chance to cool back down a little it will go ahead and work again.
The engine may be struggling to run under a load when mowing which would cause more heat. Does the engine still seems to start easy? Does it still have good power when running? You have had the blower housing off of the engine in order to replace the ignition coil, was it clean under there? No grass build up or trash in the cylinder fins.
Have you noticed the engine being any louder than normal when running? Maybe an exhaust leak causing for more heat in the engine compartment.
The engine runs good until you shut it off, right?
There has to be some reason that you are building up heat more than normal or before. When you shut the engine off it will build up heat around the engine for the first several minutes due to no air flow then it will actually start cooling back down. I believe that this heat is the reason that you can't get restarted until it has a chance to cool back down again. See what happens if you remove the hood from the machine and mow with it, shut it off and try to restart it again immediately. You could have a head gasket partially blown but I would think that you would notice a slight power loss if that was the case.