Hi Comrade,
(My god, Never thought I'd say that

) I used to work on some machinery that would develop this problem, I know where you are coming from. Similar situation with a recessed allen caphead bolt.
First off, Allen & Torx bolts are usually made of 10.8 (or harder) steel therefore cannot be drilled with standard HSS bits, Cobalt will work with care but when you have drilled it, Easy-Out's won't work as they are softer than the bolt, and just get chewed up or snap.
I found that this technique would (nearly) always work. get a short piece of 20mmx6mm (3/4x1/4) flat bar, drill a hole in it slightly smaller than the outside diameter of the bolt head. Clamp it securely over the damaged bolt head, then working through the hole, fill the bolt head with weld and subsequently weld to the flat bar. Whilst taking care due to the heat, turn the flat bar and bolt using a large shifter (adjustable spanner). I think MIG welding would work best, never tried it with a stick welder.
I'm pretty certain that as stated earlier, it is the heat that does the job.
This method only failed me once, then had to resort to the air chisel and gas axe.
The bolt in question was a high tensile M10 caphead, recessed hole in a cast iron bed plate into a blind threaded hole in an aluminium casting. The whole assembly lived in refrigerated room. A disaster waiting to happen!!