I think this problem has been discussed before so try a search.
But Benz Dr has already described below what are probably the best options for getting it out.
It’s true that freezing the tool would increase the risk of it rounding, so it might be better to freeze spray the plug instead, after heating around it. Maybe even try a couple of cycles of heating and cooling (noting that I haven’t tried those sprays, not sure if they work as well in practice as in theory). If you don’t have access to oxy-acetylene the little torch should work, as hot as you can get.
Did you apply a penetrant already? I would also give the plug a few heavy hammer blows, indirectly via a large bolt or something, before trying to loosen it.
I wouldn’t connect into the cylinder head as it might risk altering the heat distribution of the head.