F&CK an easy out. The only thing it does for you is snap off in the hole to give you a piece of tooled steel to drill. I prefer to drill the bolt with the drill size needed to tap the bolt, clean the first thread out with a pick or chisel (and time), and then chase the threads with a tap. Works every time. Of course, center drilling the bolt is critical when using this method. I reiterate, F&CK an easy out.
I see this step that you have proposed as a first step, as the very last step in the process of broken fastener removal.
First things first, heat up the area with a torch, not so hot as to damage but hot enough to break a rust / corrosion bond.
next is a 1/8" hole drilled into and THROUGH the offending fastener with a LEFT HANDED DRILL BIT. use the lefty for it may just catch and spin it out to you while drilling.
Next is a generous dose of a rust breaking agent into the hole flooding not only the hole you made but the entire cavity under it.
I then bang on the fastener with a punch, trying to jar it loose.
I'll try a shouldered remover into the small hole but gently. It may screw right out, may not....DO NOT BREAK IT OFF!!!!!! more isn't better when it comes to torque applied to hardened steel!
I'll then try a right twist bit, about 3/16" or so, sometimes that will catch and screw it farther down into the hole.
More heat, burn out those oils you put in!
More shock with a drift,
A larger remover
If it hasn't yet moved, prepare to drill and re-tap.
This is all dependent on you having drilled in the center of the bolt and not off to one side catching the threads in the female hole. You catch those threads with a remover tool, it will never ever turn!!
Finding the center of a stud is more difficult than it may appear for the exposed fastener is egg shaped with the true center being actually visually off center!
Look at the photo of my removed thread, I'm old, I was taught by my machinest father on proper drilling and removal techniques, look closely, I missed the center!
I was close enough to have it work but if I had to re-tap that hole I'd end up with a weak fastening point after the new threads were cut!
Dave