Could it be a gas cap venting problem? I did a search on that problem and sounds as if it might be similar.
It could be, but being that it is an 03 I would think if it ever had the EPA cap it was taken care of a while ago. If you do a search on gas caps here you'll read a lot about it also some pictures of what the piece looks like that should get removed. Another way to see if is your cap is when it does it and the bike has shut down, crack the cap and see if you hear air rushing in, then if it starts you'll know.
Things to check:
Gas Cap
Spark Plug wires and plugs
Petcock filter: for grit or tank liner coming apart
Clean Air Filter and Carb
Check Coil and Ignition Module
When we are new owners of a used bike, the reason a person is selling it is because they are not usually using it much any more, so they are sitting a lot and that is not good for any machine especially a Big Dog.
With my bike, I had just bought it this past January and had to trouble shoot a multitude of different problems. Few of them that also create a similar sympton but it usually occurred at any point in the ride, didn't matter if the bike was cold or hot. I had a hole in a plug wire. Then My fuel tank liner had started to come apart and it would clog the petcock filter and the cross over tubes(dual tanks), it would sputter back fire and start to die due to carb not getting enough fuel, creates a real lean condition of course. So I had to get the tanks relined. To check the liner you can try to look into the tank with a flash light, but didn't show me much until I drained the tank and removed the petcock filter, then I saw pieces of liner on it and could see all the pieces sliding around in the tank. Then I still had the cutting out when reaching temp problem that you are discribing.
The first three months I spent probably 4 hours in the garage working on it for every hour I rode. But it runs fantastic now and I really got to know my bike. And working on it for me is all part of the adventure. Although I could do with out the sitting on the side of the road deal.
IMPORTANT: Let us know what you find along the way, then this thread will be useful for the next guy.
Best of Luck!