What this man said.Drill the head out (or most of the way) with a 6-10mm bit, undo the other bolt, take the cleat off - you may have to lever it over the remains of the drilled out head. Then use pliers to unscrew the remainder of the bolt that is showing. You could faf around with screw extractors if you have them, but this is method is quicker and these bolts aren't hard to come by.
Further to this, if you drill the head off enough to remove the cleat, pull the insole out of the shoe and just remove the cleat plate from inside the shoe. Will give you more room to manoeuvre when removing the remains of the cleat bolt.What this man said.
Drill the head off, pull the cleat off, use vicegrips to unscrew the thread from the shoe. Done it tens of times without fail.
That's what I did, had to replace cleat plate as it was rusted together.Further to this, if you drill the head off enough to remove the cleat, pull the insole out of the shoe and just remove the cleat plate from inside the shoe. Will give you more room to manoeuvre when removing the remains of the cleat bolt.
Dynamite. Or C4.Bah to everything. Use a hammer. If that doesn't work, get a bigger hammer.