The Austrians call them "Marillenknödel", and the Czechs "meruňkové knedliky", but they are essentially the same in both countries.
This particular dish, despite not being the most photogenic, is one I have revered as the most delicious dessert I have ever eaten. Be aware that the measurements are purely guidelines, as this is a grandmother's recipe, not a pastry chef's recipe. Ever made quark cheese dough? It's surprisingly resilient, even when in a pot of nearly-boiling water...
500g fresh quark cheese
6-10 small apricots
Some spoonfuls of demerara sugar
Some semolina flour
Some butter - preferably soft but not melted
1 large egg yolk
2-3 tbsp granulated sugar
1 tsp pure vanilla sugar
A pinch of salt
Some plain flour, both for kneading and adding to the dough
Powder sugar for decoration
Whipped cream, sour cream and/or melted butter
Put the quark in a sieve for between 30 minutes and an hour to drain off the excess liquid. Put it into a bowl and add the soft butter, egg yolk, semolina flour, sugar and vanilla sugar then mix it all in together with a wooden spoon, but don't mix it in too well as it needs to keep its rough, unpretentious nature. Add some ordinary flour at this point and mix it into a fluffy, light yet stick dough.
At this point, you will need a lot of flour on your hands, and on the board you are using. Better a ceramic one, as this can get really, really messy.
Using the wooden spoon, put the dough on the board and roll out into a log, then cut off a slice.
Flatten it using your fingers, then take an apricot (usually whole, pipped and with a slit wide enough to slip some demerara sugar into it, but in my case, I quartered a half-slice and sprinkled some on top) and enclose the dough around it. Roll them into balls with your hands.
It is almost impossible, with such a volatile dough, to get it totally enclosed, but it's not too important. The stuff is very resilient. Put them on a plate ready for boiling.
Bring a saucepan of water almost to a boil, and put some dumplings directly in it using a perforated serving spoon. Make sure they don't stick to the bottom of the saucepan, and check regularly. As soon as they rise to the surface, they are ready.
Put them onto a plate, pour the melted butter on top, sprinkle some powder sugar if you want, and serve with whipped cream or sour cream. If it's a hot day, ice cream goes down really well.
Here is a photo of a dumpling that has been opened: