First off, I'm German, so you would think I know, but it seems traditional cuisine has not been passed down my family tree.
This question really consists of two parts:
- What makes potatos dough (or I guess, starchy dough in general, there's all kinds of dumplings, there's pasta, etc) keep its shape when you boil it in water? This may extend to some degree to deep frying.
- What properties make balls of dough "fluffy" or "textured", but not "tough", "rubber-like" etc.? I guess this will go somewhere along the lines of what structure the starch granules form with the water, how stable it is, and how dense.
There is about a million recipes online for cooked, raw, and 50-50 knödels. Some have egg, some don't, and just about everywhere there's people asking how to do them right. There's also a question to that here on this site.
However, it's hard to get some actual information on the key aspects (this is a problem I have with recipes in general). I'm pretty sure there ought to be no egg and not a lot of flour in the dumplings. I'm also certain that either all cooked, all raw or half-half are common and widespread varieties, but there it stops.
I would like to find some science based instructions, the kind of articles Kenji runs on The Food Lab. It looks like most recipes have some way of enriching the dough with starch above what's in the potatoes anyway, by draining water from the potatoes and adding starch powder. The father of a friend wraps the potatoes in cloths and puts them through the spin cycle of the washing machine, which (because of the preceding meticulous cleaning of the latter) is a tedious procedure that's only done for Christmas.
Millions of grandmas have reached the proper end result via various routes, using plain ingredients and unsophisticated equipment, so there should be some basic principle at work, which can be analyzed with science, very much like with roasting beef or frying potatoes.
Maybe, we could dig up something about related recipes, like gnocchi, which would be helpful.
Best Answer
Your dealing with a few things here. First is starches in general. The thing to know about starches is how they gelatinize and at what temperatures.
This powerpoint is a nice primer on that topic. www.cfs.purdue.edu/class/f&n630/gelatinization.ppt
Basically your dealing with amylose and amylopectin, together they are what we know as a starch. When they come in contact with water the starch cells begin to swell and when their gelatinization temperature is met they burst and release their contents into the medium they are in. In the case of a dough ball, your dealing with tons and tons of little cells being held together loosely at first by the physical pressure of kneading them into balls and then when the heat causes the starches to gelatinize they adhere to one another kind of like being caught in a net.
Now the fluffiness portion of the question can really depend on how the dough is being cooked. For something like a dumpling being cooked in the boiling water, one would typically want to work the dough as little as possible to avoid making it too dense and if using flour to lessen the gluten formation that can make it very chewy like bread. Some recipes call for leaveners that can create gas bubbles when heated to a certain temperature and then through gelatinization the bubbles are trapped inside the dough and create an airy texture.