If we blow up a global killer (in the 3 - 20km range) days before impact into enough pieces that none of the them exceeds, let's say, a 30m diameter and the impact pattern is dispersed over most of the athmosphere, would that be a possibility that the impact energy is soaked up with less catastrophic consequences than the hit of the asteroid in one piece? Or would the distributed energy heat up the athmosphere so much that it would kill higher life instantly or eventually over chemical/biological chain reactions? Or would the early impacts heat up or disturb the athmosphere so much that later (in the range of seconds or minutes) impacts would experience not enough friction so that there would be a surface bombardement with millions of little pieces?
I don't think it's doable to pulverize a dino-killer on such short notice. But years of movie and TV sci fi have stretched my WSOD bubble enough that I can consider this.
For an 3 km diameter asteroid moving 13 km/s and having a density of 3 tonnes per cubic meter, I get kinetic energy of about 3.6e21 joules.
If my arithmetic's right, earth receives about 1.75e17 joules of sunlight each second.
So my BOTE seems to indicate about the same energy as 6 hours of sunlight.
A hemisphere receiving the energy equivalent of 6 hours of sunlight over a brief time -- it wouldn't be pleasant.
I believe we'd be screwed whether the impactor was one chunk or a cloud of gravel.
TL;DR version: Too big and way, way too late.
The dispersal can't be done, even at the lower end of that 3-20 km scale. Holsapple claims 5 kilojoule/kg are needed to disrupt and disperse a solid 1 km asteroid asteroid, with energy scaling with radius1.65. Disrupting and dispersing a solid 3 km diameter asteroid with a density of 3 g/cc would require a deeply buried two Tsar Bomba weapon. The energy needed to disrupt and disperse a 20 km diameter solid asteroid is beyond anything humanity can achieve.
This disruption and dispersal will not be uniform. It will instead create a few big chunks, perhaps 1/3 the diameter of the original body, a larger number of intermediate sized chunks, and thousands upon thousands of little chunks. Those big chunks are still civilization killers. The intermediate-sized chunks will make Tunguska look small, and the thousands upon thousands of little chunks -- we recently saw what a little chunk can do, over Chelyabinsk.
What if it's a rubble pile or a comet? While they're not quite as dense, they need much, much more energy to disrupt and disperse. Rubble piles and comets are very good at absorbing impact energy. Disrupting and dispersing a one km diameter comet is beyond the scope of human technology.
The time scale, two days, is extremely short. A decade of advanced notice is considered to be very short notice when it comes to diverting a one km impactor. Even larger ones require even more advance notice. A century's advance notice might be enough time to deal with a 20 km diameter object.