Prepare Cabbage: Melt 3 tablespoons of butter in a large skillet over medium heat. Add 1 chopped large onion and sauté for 3-4 minutes until soft. Add 4 cups of shredded cabbage, season with salt and pepper, and cook for 10-12 minutes until softened. Pour in 1 cup of broth, cover, and let simmer for another 10 minutes.

Make Dumpling Batter: In a mixing bowl, whisk together 2 cups of flour and 1 teaspoon salt. Make a well and add 2 beaten eggs, 1 cup of milk, and 2 tablespoons of melted butter. Stir until thick and slightly sticky. Let rest for 15 minutes.

Cook Dumplings: Bring salted water to a boil, then reduce heat and drop spoonfuls of batter into the water. Cook for 3-4 minutes until they float. Use a slotted spoon to remove and set aside.

Combine Everything: Gently transfer dumplings into the skillet with cabbage. Toss carefully and sauté for another 5 minutes. Drizzle with vinegar if desired.

Serve Warm: Ladle portions onto plates and garnish with fresh parsley if desired.