Since I’ve had this problem twice, I’m sure there’s more than one person out there who had this problem too. Disclaimer: I’m not a professional, and my advice is anecdotal.
If you want to see the broken bolt before I start on it, look at my post on the ENO Hub.
First thing I like to do is take a die grinder (or a dremel with a metal bit) and smooth the surface into a conical shape which makes it easier to center the cutting tool when I get started.
This is the tool you need to get the job done. One end is a cutting tool and the other is like a screw with reverse threads. Slowly drill a hole in the center of the bolt and use the other end to back it out in reverse. I have the Sears version, and the package has complete instructions in it.
A little cutting oil and some slow drilling, and that’s what it looks like after 15 min or so. You have to be reallllly careful with this step because if you break the bit (which is made of hardened tool steel) off in the center of the bolt then it will be all but impossible to remove. All that’s left is to reverse the bit, and back out the bolt. Letting a little penetrating oil soak into the threads at this point will help it go easier if it’s seized up.
And there you have it. Easy enough.