Finding Anesthesia Specialist

The only reason you would need those specialists are when you need them for research purposes. You don't need to be able to physically pick them out in your base, just having them somewhere in your research team should be enough.

First, you can determine if they are even in your staff by looking at the mission that contained him and look at the Obtainable Skills/Blueprints section. If the skill is dimmed out, you have someone who has that role on your staff, otherwise it will be white.

I only have a Tough Guy, but no Anesthesia, Mechatronics, or Transportation Specialist.

Once you are certain he's in your staff, you can go to Staff Management and Auto Assign your staff and take a quick look at your research team. Or if you had recently captured/recovered him, check your sickbay/brig. If you cannot find him immediately, you'll have to search through all of your staff.

Under the All Staff view, sort your staff by Skills (descending) and scroll over to the guys with wrenches. They'll appear near the top of your staff grouped together. Make sure you're in the display where you can view all your staff's skill and ability levels.

Found the guys with the wrenches.

Then when you've found the wrenches, change the display again to view the individual's stats and scroll through til you find your man.

Found him.

He might've been automatically dismissed if you reached maximum capacity, if that's the case you'll have to hire him again in mission 10.

Otherwise, you can see your full list of soldiers, sort them by special ability, and look through the ones with R&D skills.