How can I make nutritional yeast?
I have to humbly disagree with SAJ14SAJ (I still have the utmost respect given the breadth of knowledge). Most of the steps you need to perform are regularly used by homebrewers and hobby mycologists to renew or isolate special strains of yeast/fungi that are otherwise unavailable. For instance, if you wanted to isolate the yeast used by Chimay to brew the same style of beer using the same yeast as the brewery, your best option is to culture the yeast from one of their beers. The only major exception with what you would like to do is the addition of drying at the end. 

While safety cannot be guaranteed when doing this outside of the lab, following decent sanitary procedures it is absolutely possible to do at home. Some specialized equipment IS required but can be found at your local restaurant store, amazon, or a lab supply store. Basically, be clean, be sanitary, and don't be dumb and you should be fine.

There are 4 essential steps you will need to perform.


Create a sterile substrate for the yeast to multiply in
Isolate yeast cells and innoculate the sterile substrate
Wait for the yeast colony to grow until it flocculates (falls to the bottom of the solution)
Wash and dry the yeast


First, is create a sterile substrate. You can follow the steps here:
https://byo.com/stories/item/434-canning-yeast-starters OR
http://www.maltosefalcons.com/tech/starter-made-easy-pressure-cooking-your-starter-wort-ahead-time

I do disagree with their using a hot water canner in this situation. Wort can be either acidic or alkaline, so there is no sure way to know that botulinum spores have been eliminated. I would suggest using a pressure cooker at 15PSI for 15-20 minutes. Once they are cooled, they should keep for up to 1 year.

Next, take a small sample of whatever yeast you are working from (almost definitely s. cerevisiae) and use it to inoculate the cooled starter. The best practice here would be to wash the yeast first, (follow this guide: http://billybrew.com/yeast-washing). Then, mix the resulting washed yeast with a little distilled water, suck into a sterile syringe, and innoculate the starter. At this point, you want to close and shake the starter like your life depended on it. An even better option would be to use a flask on a stirplate with a stir bar. You want the jar to be protected from foreign bodies but still allow air exchange. You can modify the lid by installing an airlock, or if you really want to be avant garde, just remove the lid and use a rubber band to seal the top of the lid with tyvek. Unlike if you were trying to make beer, oxygenation isn't an issue here so the one-way functionality of a normal airlock isn't really necessary. Set it in a cool dark location at around 70 degrees F.

Let the yeast do their thing. After a few days of fermenting, you will see the yeast fall to the bottom of the jar. This is a natural process of yeast called flocculation - the small yeast particles tend to cling together, and eventually they will form large enough clumps that they will fall out of solution. The layers that form on the bottom of the vessel are called "trub" in brewing. It's a mixture of yeast and other undissolved particulates that have fallen out of solution. Once you see this has happened, you can move on to washing the new yeast using the same washing procedure as before. If at this stage you sense any off-smells, throw it out and start over - trust your nose and your eyes here. It should smell yeasty and bready. It shouldn't smell bacterial or sick-sweet.

Once you have the cake of washed yeast, you can place it on a piece of STERILE aluminum foil and place it in a food dehydrator with heat. It should dry into a large cake. This can then be crumbled down to be used as nutritional yeast. The heat should be set at 125-130 degrees, which should critically injure/kill the yeast, deactivating it.

The whole process will take a few days, but all of the steps above are done at home by homebrewers and hobby mycologists currently - there's no reason you couldn't do it yourself. Just make sure you keep everything clean and sterile as much as possible. If it seems like too much of a hassle, I'd look out for mail order sources of the yeast. I will warn you though, that this might not produce a ton of yeast without scaling things up a bit (which is totally doable).

Also in response to some other comments - bakers yeast, brewers yeast - they're all different strains of s. cervisiae. So, same species, different strains. The different strains are specialized at certain things, but can be substituted for one another in a pinch (the results won't be as great, but they'll work). Remember, baking was a byproduct of brewing as far back as Egyptian times, possibly earlier. We've been using s. cervisiae almost as long as we've had civilization. Possibly longer.

Also, fun fact - you follow more or less the same steps any time you are working with fungiculture (growth of fungi). This makes sense considering yeast ARE fungi. With a mushroom, you would take a thick sterile syringe to act as a punch, take a biopsy sample from the stem (pierce from one side clear to the other and then eject the sample), cut off the sides of the sample that were exposed to the outside world, and using the remaining portion the same way you'd use the yeast above to get a mycelial culture. The only difference is instead of using a malt starter like yeast, you use agar.