You have a mile wide asteroid coming in. You nuke it. Then you have four quarter miles asteroids coming in hundreds if not thousands of mile apart.
That could work!
/s
I’m more inclined to think we need to develop something that could meet the asteroid millions of miles from us, that would exert thrust from the side to change it’s path.
Doing so far away, we would probably only have to change it’s path a fraction of a degree.
Not splintering it, we could avoid all the debris that would accompany a nuclear detonation. And we wouldn’t be able to guide that debris in any direction.
We could be developing a space shotgun shell to hit earth in many locations with massive space debris.
Even reducing (speculatively ) a mile rock to 4 quarters would be significant. One of your primary objectives will be to prevent impact of an object capable of punching through the lithosphere into the asthenosphere. Punching through to that will be bad. The size threshold for that currently is probably about 6-10 miles depending on composition and velocity.