Cables are going to have to be used. Astrnauts have to have them as guide ropes, or to brace themselves so as to apply force without knocking themselves into space. Cables also cut up the Kursk submarine you know. A solid nickel-iron slug cut be cut similarly.
But here is the capper. Many asteroids--all of them probably--rotate/tumble. What cutting an asteroid does is turn it into a bola. Release one end of it at a certain time when facing a certain direction, and you convert rotational forces into translational ones--reducing engine/fuel costs.
Call it David's sling