At the prompting of Rex Roof I thought I'd post the general method I've been using for making ginger beer. I'm now on my fourth batch, and already the method has varied quiet a bit from something roughly based on the ginger episode of Good Eats to something roughly based on the ginger beer recipe in Wild Fermentation by Sandor Ellix Katx.
This method makes four two-liter bottles of ginger beer. It can be scaled. First, completely and thouroughly clean four empty two-liter bottles, and make sure they are completely rinsed out.
In a large pot pour four cups of sugar and four liters of good water. Start to boil. While that is happening, take one pound of ginger root (the older stuff, with the thicker and darker skin, is more pungent) and grate it well. Grate on something that will catch any juice that comes out of it (I use a tray). Once the water has reached a boil, add the ginger and any juice that came out of it. Keep at a low boil for fifteen minutes then turn off the heat. Zest two lemons, add that to the mixture, cover, and let be for one hour.
After the hour is up, squeeze the juice of those two lemons into the mixture, and then filter the ginger solids out of the water. I find this easiest to do if you take another large pot, put a metal cooling rack on top of it, sit a metal mesh collander on that, and line the collander with a few layers of cheesecloth. Carefully pour the liquid through this — I find it a good idea to do this all over that big tray, to catch any spills. Once you've poured everything through the collander, use the back of a ladle or something similar to squeeze as much liquid out of the solids as possible &mdash do not be tempted to do this with your hands, or by squeezing the cheesecloth, it's very hot at this point.
Pour all the liquid back into the large pot. Add another four liters of water. Use a thermometer that you trust to take the temperature of the water, we want it to be around 95 deg. F. Take about a cup of this liquid and put it in a glass or something, add 1/4 teaspoon of bread yeast and stir well. We're proofing the yeast at this point, so just let it sit. In the meantime, pour the liquid into the two-liter bottles. A ladle and funnel is best for this. I usually fill the bottles just to the point where the top starts curving in (this leaves some head room for gas to accumulate, which I think aids in carbonation). If you have any liquid left over, put it in the fridge for later. After that's done and a couple minutes have passed, look at the yeast-liquid mixture. Does it smell good and yeasty, and have some bubbles? If so, you have good yeast. If not, you need different yeast, come back when you have some.
Divide up the yeast mixture evenly between the four bottles. Cap the bottles, shake them well to make sure the yeast mixture is evenly distributed. Now we need to keep the bottles warm. I think the wee beasties that are turning your sugar into carbon dioxide and carbonating your beer are happiest between around 85 and 95 deg. F. What I like to do is take a large chest cooler, put the bottles in and fill the cooler about 3/4 of the way up the side of the bottles with 90 deg F. water. Kinda like a water bath. With the cooler lid closed, you have a fairly stable temperature environment. If it's warm enough where you're at, you can keep them in a quiet spot somewhere. And, even if it's a bit colder you can still keep them out, it will just take longer for the carbonation to happen. There is an advantage to the cooler though: if for some reason your carbonation is very active and a bottle ruptures, the cooler acts as a containment device.
Personally, with the mixture in 90 deg. F. water and kept in the cooler, I let it ferment for 48 hours. After 24 hours, I carefully open the bottles to let out a bit of gas (the bottles are usually pretty gassy at this point, and I would rather not have one go off, even if it is contained). NOTE: if you happened to put some stuff in the bottle to add flavor while it is sitting (like, say, a stick of cinnamon), those ingredients will make an INCREDIBLY active nucleation site. Open the bottles very very slowly. Your nose, and half of your kitchen, will thank you for heeding this advice. More on flavoring later.
Okay, 48 hours are up. If you look at the bottles you'll see some stuff at the bottom. This is basically dead yeast that have settled there. What I like to do is open the bottles (very slowly) and pour them, again, slowly, through a few layers of cheesecloth, trying to disturb the stuff as little as possible and generally filter it out. If you added things to steep in the brew while it was fermenting, this also clears that out. Rinse the bottles, pour the stuff back in, and put the bottles in the fridge. If you have any of the leftover un-fermented mixture, you can use that to top up the bottles, or boil 4.5 cups of water with 0.5 cups of sugar for a few minutes, and add that. From my experience, about three days in the fridge and this stuff starts tasting really good.
Now, what we've done here is used yeast to turn sugar into alcohol and carbon dioxide. Since the process in ginger beer is really about carbonation we're aiming to get enough fermentation to do that but not go all the way into turning all of the sugar into alcohol (at least, I'm not. Feel free to go forward if you want). Note, however, that there will be at least a trace amount of alcohol in the ginger beer, although this is usually fractions of a percent. Storing the finished product in the fridge will slow down considerably the yeast, it will not stop them. So, if you forget a bottle and open it up a couple weeks from now, it's probably going to have quite a kick. Also, burp the bottle once or so a day, so you don't get ginger beer exploding all over your fridge.
Flavorings: this last batch I've experemented with some flavorings to the ginger beer. In one, I added a stick of cinnamon during fermentation. In another, I added 1/4 of a cup of honey and mixed well, after fermentation. I have a suspicion that honey might affect the fermentation, which is why I added it afterwards (then again, mead is essentially fermented honey water, so maybe you don't need to do this). I've also added some other stuff to one bottle, but that's my little secret for now. I may want to add more lemon juice, I'm not sure. In general, one pound of ginger for four two-liter bottles seems to give the right amount of kick (I like it strong, but not so strong you can't enjoy it. But strong enough indeed!)