Also, there’s beam divergence (https://en.wikipedia.org/wiki/Beam_divergence). That has a theoretical lower limit (http://www.aml.engineering.columbia.edu/ntm/level1/ch02/html...). Reading https://www.reddit.com/r/askscience/comments/1ihca9/how_much..., that limit is about 1:3000, so a laser beam is at least 100 km wide at the earth-moon distance.
I guess you would want to start rotating that asteroid way earlier, and that you would want to hit asteroids smaller than that. That makes it practically infeasible (Given infinitely precise aim, you theoretically still would be able to hit it only at one side, by aiming off-target, but that would wasting most of your beam energy)
Also, I am not convinced evaporating the surface by heating it from an angle would significantly impart angular momentum.
Always surprised how little science there is in this stuff, both detection and diversion it is a real existential threat to humanity given enough time and yet militaries around the world are out there tilting at windmills spending all their cash on outdated bloated acquisition projects to kill each other better in the most peaceful time on Earth.
I tried to figure out just how accurately we are able to point stuff in space and found out the hubble telescope is able to "lock onto a target without deviating more than 7/1000th of an arcsecond, or about the width of a human hair seen at a distance of 1 mile." [1]
That's pretty amazing to me. Though I'm not sure if that would be enough, or whether we can be even more accurate.
With a laser beam it might also not be a problem if you're just slightly inaccurate, say only hitting your mark 70% of the time as long as you still manage to heat your target spot up sufficiently.
[1]: https://www.nasa.gov/mission_pages/hubble/story/index.html
Since people smarter than me have floated this as a way of deflecting asteroids, I assume there's some kind of way to focus it over those distances? It seems nontrivial though.