Peel the potatoes and cut them into cubes about 2 cm in size. Peel and finely chop the onion.
Pour water into the pot, add salt. Add the caraway seeds, pepper, a tablespoon of butter, the diced potatoes, and onion. Bring to a boil.
Pour the milk into a bowl and add a ladleful of hot water. Add the flour to the lukewarm milk-water mixture and whisk thoroughly.
Remove the pot with the simmering water from the heat. Slowly pour in the flour whisked into the milk, and continue whisking until the sauce is smooth.
Put the pot back on the stove, turn the heat down to low, half cover with a lid and let bubble for 15-20 minutes, or until the potatoes are soft.
In another bowl, whisk the whole eggs.
Set the pot with the sauce aside; we will not cook it anymore. Now gradually pour the egg mixture into the hot sauce, mix well. The egg will soften the sauce, increase its richness, and thicken it even more.
Season with salt and pepper, if necessary, and serve.