Melt butter in a large pot over medium heat. Sauté onion until soft and translucent.
Add chopped cabbage, season with salt and pepper. Cook until cabbage is tender, about 20 minutes.
Whisk flour and salt in a medium bowl. Beat eggs and milk in a separate bowl. Gradually add wet mixture to dry ingredients, stirring until a thick batter forms.

Bring a pot of salted water to a gentle boil. Drop small portions of dumpling batter into boiling water using two spoons. Cook until dumplings rise to the surface and are cooked through, about 3-5 minutes. Remove with a slotted spoon and set aside.

Fold cooked dumplings into tender cabbage.
Garnish with chopped parsley if desired. Serve warm.