Whoever is capable of solving the real challenge of delivering the warhead.

Extrapolating from the existing high-yield bombs, the mass of a 1-gigaton warhead could be something like 200 tons. The Starship would probably be large enough to launch it into orbit, if you integrate the warhead in the upper stage and discard the lower stage. Then you would need 10+ refueling flights before you can launch it towards the asteroid.

A 10-gigaton bomb would need a rocket 10x bigger than the Starship. Or you would have to build it in the orbit.

Ivy Mike was around 70 tons and yielded 10 Mt, the B41 was around 5 tons yielding 25 Mt.

Extrapolating from that a 10 Gt device should weigh around negative 25 thousand tons.

You wouldn't need the starship to launch it, just attach it to the HMS Invincible and it would spontaneously float upwards into space.

Not sure, but it would be deployed by a ragtag team of oil rig miners...