1 gallon brews are my MO at the moment. I'm packing up for a cross-country move and will jump into bigger batches afterwards.
I use a 12qt pot as well. Go to Home Depot or Lowes and get 5 gallon paint strainers. Use them for Brew in a Bag, which means you can mash, lauter and boil in one pot.
You won't need two gallons of wort. Here's how my brew day goes:
1) Calculate the water/grist ratio I want to go with (usually 1.25 qt/lb). A one gallon stout with an average gravity will be about 2.5-3.5lbs of grains. Absorption rate is .1 gallons of water per pound of grain. I'll go middle of the road and use 3 lbs of grain for this example. The total strike water for 1.25qt/lb for 3lbs of grain is 3.75qts of water, or almost a full gallon. However, the grain's water absorption will significantly decrease your ratio if you don't account for it. 3lbs of grain will absorb .3 gallons of water. So, add the two to get your strike water amount. 3.75+.3=4.05qts.
2) Calculate your strike water temp. With one gallon, the temp is pretty easy to manipulate. I usually heat my strike water to ten degrees above my target temp. Put the paint strainer in the pot and roll the top elastic band around the top of the pot. Now, you have a filter that can separate your grain from your wort after the mash. After mashing in, you can leave the lid off and stir if the temp is too high, or you can turn the burner on low and heat it up a few degrees. When you have mashed in and are satisfied with your temp, put a few hand towels over the top of the pot (sort of draped over the edges) to help hold the heat in.
3) While the mash is going, I usually just prep for the lauter and boil. When you have about 15 minutes left on the mash, start heating up a quart of water to about 168 for the sparge.
4) When you're done mashing, pick the grain bag up and let it drip all of the wort out. When dripping slows down, put a strainer over the top of the pot and set the grain bag in it to finish dripping. My strainer has handles on the side that sit on the lip of the pot. Turn your burner back on and start heating up the wort for the boil.
5) You can do this step by yourself, but I prefer to ask my wife for help. I usually hold the grain bag up and open, and then my wife pours the sparge water over the grains. Then, I let it drain again. If you don't have someone to hold it for you, no worries. Pour the sparge water over the bag, then stir the grains around to get out extra wort. Discard your grains.
6) When the wort is heating up, rehydrate your yeast, if using dry, so it'll be ready when you're done. I don't use liquid yeast and personally don't care to. Dry yeast is convenient and there are many options. Do your boil and follow your hop schedule. A few minutes before flameout, fill your sink with cold water and stir in a cup of salt. When you turn your burner off, move the kettle to the sink and dump a lot of ice in it. Your beer will go from boiling temp to pitching temp fairly quickly, usually around 10 or 15 minutes.
7) Because of the quick cooling, a lot of break and hop material will cone up in the center of the kettle. So, you can try siphoning from the edge to get clear wort, but I usually don't worry about it and just dump the whole thing in, or run it through a different grain bag to strain it out. Pour the rehydrated yeast into your carboy and put your wort in on top of it.
8) Once your carboy is full (don't go higher than the bottom of the neck, it'll cause a mess), put a cork on it and shake the holy hell out of it. It will introduce a little bit of oxygen into the wort, albeit not a huge amount. I usually put a napkin or rag over the top of the carboy (with a rubberband to hold it in place) for the first few days because the yeast need the oxygen and they spew a lot of CO2. When I do that vice putting an airlock on it right off the bat, I find that my fermentations are way more healthy and I get better attenuation.
My batches usually ferment out pretty quickly, 2-4 days max. Sorry for the crash course. If you have any questions, feel free to PM me. I've gotten quite good at one gallon batches. I think they are great for trying new recipes instead of making 5 gallons of undrinkable beer. When I finish my move, I'll move to five gallon batches to fill corny kegs and I'll use one gallon batches for one-offs or test batches.
