Peel 4 medium potatoes and cut them into 1-inch (2.5 cm) cubes. Peel and finely chop 1 medium onion.
Pour 4 cups water into a pot and season with 2 teaspoons salt, ½ teaspoon caraway seeds, ¼ teaspoon ground black pepper, and 1 Teaspoon unsalted butter. Add the cubed potatoes and chopped onions, then bring to a boil.
Pour 1 cup milk into a bowl and add a ladleful of hot water from boiling potatoes. Add ⅔ cup all-purpose flour to the lukewarm milk-water mixture and whisk thoroughly until smooth.
Remove the pot with the simmering water from the heat. Slowly pour in the milk with whisked flour, continuously whisking until the sauce is smooth.
Put the pot back on the stove, reduce the heat to low, half cover with a lid, and let it simmer for 15-20 minutes, or until the potatoes are soft.
In another bowl, whisk 3 medium eggs.
Set the pot of sauce aside; you will not cook it anymore. Gradually pour the egg mixture into the hot sauce, mixing well. The egg will enhance the sauce's richness, soften its texture, and thicken it even more.
If necessary, season with salt and pepper to taste, and serve.