There could no doubt be targeted government recruiting of people with certain expertise, as I assume there was in the past among nuclear or rocket scientists. This is not the same as a draft, it's driven by special skills, not a broad need for manpower.
How has the military (US, or other) historically managed to recruit and retain their medical corps, or the Army Corps of Engineers? Has there just not been a situation in modern memory where there weren't doctors sufficient to keep the gears of war turning?
I've worked with quite a few Federal and DOD "hacker" types over the years in the capacity of a hacker in the private sector. The impression that I have is such that your average ruck-humping Joe will not be able to do it, but it's not so special (as contrasted with a civil engineer or doctor) that you couldn't take high-aptitude recruits and bootstrap their knowledge to a sufficient operational capacity in the amount of time as a typical post-boot AIT timeframe as you might a Corpsman or linguist.
Our not-quite-aptly named SOC would routinely engage in legally grey operations to disable cyber-criminals servers whose hosting providers were non-responsive to our standard, legal course of take down. The majority of the staff had a Bachelors in an unrelated domain, several with an associates in general IT work, and a few with nothing more than a programming bootcamp.
The tale that we're so special as to not be meaningfully draftable is, I feel, egotistical.
The US military will pay for someone's medical school education, and also offers significant signing bonuses for doctors, who then enter the military as officers. Which I imagine is enough to keep a steady pipeline in regular times.
These days so much of US military work is done by contractors, that I imagine they'd just pay contractors whatever was necessary to attract what they needed for specialized non-combat roles. (Heck, contractors end up being in effectively combat roles these days too, even if formally not. Wikipedia says as of 2012, 1,569 US contractors had been killed in Iraq, compare to ~4500 soldiers).
Back when there was a draft in the US, they drafted doctors along with everyone else, and had them be doctors in the military. (Some of my family members were such in WWII). I think they just had draft numbers that came up like everyone else's though, I don't think there was a specific doctor draft.
The problem I saw when I was in was that the retention problem was one of incentives. Enlisted folks who work on offensive cyber are sitting next to people who are making far more than they are for nearly the exact same job, minus the aspects of being a soldier (long hours, no OT, tasks outside your core job). Being a contractor just pays more for less, so they have little reason to stay in when the promise of a better life isn't quite what their recruiter told them.
I find it interesting that you've used the exact word that they use, "special skills draft".
I was on the draft board in my county during the Iraq war. To be clear-- the draft board was never activated, but I had found out that you could apply to be on it, and I was selected because basically nobody knew about it or wanted to do it.
The reason I wanted to be on it, was there was lots of talk about a "special skills" draft for nurses and technical people. The idea was they could slip this one under the radar because the bulk of Americans wouldn't be concerned. Then it becomes much easier to expand the draft.
I reasoned that if I was ON the draft board I couldn't get drafted, and that I could try to treat people compassionately.
Some MOSs are scarcer though. Cyber is in a sense a form of special operations because it requires elite uncommon skills applied in war through unconventional means requiring exceptional enabling personnel and tools.
But if you try to give those same 1,000 people a software job, most of them would be terrible. Hell, most people employed as software engineers seem kind of bad at it. It isn't the type of job that lends itself to a draft.
Of course, you aren't limited to just the first 1,000 people you see. Maybe you ask FAANG for a list of their engineers, and randomly take 10% of them. Good goddamned luck with that. Google has had trouble getting their own engineering teams on board with DOD work. You try to draft those engineers? You will see a vindictive pettiness unmatched in human history. It would be the least productive engineering team ever assembled.
[1] Well, that used to be true. The general health of the American public isn't great right now, either.