Stir the warm milk, sugar, and yeast together in a large bowl. Let the mixture stand for 5–10 minutes, until foamy.

Mix in the melted butter, eggs, and salt. Add the flour gradually until a soft dough forms, then knead for 6–8 minutes by hand or with a dough hook until smooth and elastic.

Place the dough in a greased bowl, cover, and let it rise in a warm place until doubled, about 1 hour.

Brown the ground beef and onion in a large skillet over medium heat, breaking the beef into small pieces. Drain excess fat if necessary.

Add the shredded cabbage, salt, and black pepper. Cook for 10–15 minutes, stirring occasionally, until the cabbage is tender and most of its moisture has evaporated, then cool the filling to lukewarm or room temperature.

Punch down the dough and divide it into 12 equal pieces. Roll each piece into a roughly 6- to 7-inch square or circle.

Place about 1/2 to 2/3 cup of filling in the center of each dough piece. Pull the edges over the filling and pinch firmly to seal, then place each bun seam-side down on a parchment-lined baking sheet.

Cover the filled buns loosely and let them rest for 20–30 minutes. Meanwhile, heat the oven to 350°F.

Bake for 18–22 minutes, until the tops are golden brown. Brush the hot runzas with melted butter for a softer crust and richer flavor, then serve hot.