However, thanks to Justin Taylor Tate's recent Instructable for Cepelinai (which are also both gluten-free and delicious), I now know the key to making my gnocchi stick together! Thank you, Justin!
I'll give you the spoiler-alert version now: The trick is to get as much of the water out of the potato dough as you can. Once it's dry enough, it will stick together when you boil it, even without adding eggs or gluten-like substances. In fact, you can make pure-potato gnocchi if you want to.
2 to 4 large potatoes
A dash of lemon juice or other acid (optional; keeps the potatoes from discolouring)
1 egg (optional)
1/4 tsp salt or to taste
Plus: Whatever toppings you want to put on your gnocchi, e.g. butter and sour cream, or tomato sauce and cheese, or mushroom, cream, and bacon fat sauce.
A blender or a food processor (optional; I'll explain below.)
A large pot to boil water in
A large bowl to act as a basin to catch potato water
A piece of cheese cloth or a clean tea towel you're willing to use as cheesecloth
Wash and peel your potatoes. (If you want to keep the skins on, that works too, however. Easier and healthier!)
If you're using a blender, dice your raw potatoes into approximately 1 cm (0.5 inch) cubes, i.e. small enough to not choke your blender.
Throw the potatoes into the blender or food processor. Add the lemon or acid (this keeps the potatoes from discolouring.) Now turn it all to mush. If the potatoes don't want to blend, feel free to pour in half a cup of water to help the blender chew on them. You'll be squeezing all that water out later, anyhow.
NOTE: If you don't have a blender or food processor, then bake or boil your potatoes until they're soft, and mash them. Proceed to the next step once they're cool enough for you to handle.
Now it's time to squeeeeeeeeze your potato mash to get the water out. Pull the edges of the cloth together so you have the potato mash confined in a ball inside the cloth. Now start squeezing and wringing it.This takes some time and a bit of muscle power. Think of it as the workout-equivalent of kneading bread dough, except for gluten-intolerant people. :)
Get as much of the water out as you reasonably can. If you're desperate, you might try putting the ball of dough between two cookie sheets or plates and then sitting on it, but I can't be held responsible for any plates you break or any mess you make!
When you're done, the dough should be quite dry and a little crumbly. You can squeeze it together into clumps in your hand. It will almost seem too crumbly, as if it couldn't possibly hold together once it hits the boiling water. Trust me, this is the right consistency.
Justin suggested straining the potato water to recover the starch that was lost with it. I just waited a few minutes, poured off the liquid, and scooped the goo that had settled to the bottom of the bowl out and into my potato dough.
If it starts to dissolve, that means you didn't get enough of the water out of the potato dough. You can either try squeezing your dough some more, or you can add some gluten-free flour to it, keeping in mind the only reason you're adding the flour is to absorb some of the water in the potato dough.
However, if it didn't fall apart, then you're ready to finish preparing your gnocchi! Add the salt and the egg, and mix everything together well. Form the gnocchi into bite-sized pieces. (That's optional: You can also just drop bite-sized pieces into the boiling water with a spoon.)
Throw your gnocchi into the boiling water, taking care to not over-load the pot (you don't want the water to stop boiling.) When the gnocchi are persistently floating at the top of the water, they're finished cooking.
Strain your gnocchi, add whatever toppings you want, and serve! (But read my warning about serving size in the next step.)
They DO taste quite a bit like pierogies, so butter, sour cream, and fried onions are a good mix with them. So are the more traditional Italian sauces and cheeses. Justin also mentioned that cepelinai are often served with a sauce made from cream, mushrooms, and bacon fat--and that sounds like it'd be good on gnocchi too!
Like pierogies, gnocchi can be frozen when they're raw to provide you with fast meals later. Freeze the bite-sized dollops on a cookie sheet, then put them in a ziplock when they're hard. You can throw them in the boiling water frozen when it's time to make your meal.
Also like pierogies, if you have leftovers (ha!), you can put them in the fridge and fry them up the next morning for a crunchy/hearty breakfast side dish.
Please note the recipe quantities I listed in Step 1 are very approximate, and constrained by how many diced potatoes I could fit in my blender. Feel free to experiment with the quantities. You can incorporate more egg or no egg, or add a flour of your choice to the dough.
Enjoy! (And thanks again to Justin for providing me with the key to making great gluten-free gnocchi!)