Prepare cabbage head by removing the stalk. Cook head in a large pot of boiling, salted water for about 10 minutes to separate leaves. Remove leaves and set aside to cool.
Once the leaves have cooled, turn each leaf over so that the thick vein, or hard rib, is on top. Trim hard rib with a sharp small kitchen knife.
Pat leaves with a paper kitchen towel to remove any excess water.
Mix ground beef, ground pork, eggs, onion, breadcrumbs, salt, steak spice, and pepper together in a large bowl.
Place meat mixture, about 1 heaping tablespoon, on the bottom end of each cabbage leaf. Roll up tightly, tucking in sides. Fasten the stuffed cabbage rolls with a wooden toothpick.
In a large saucepan, heat butter and oil over high heat. In batches, carefully brown cabbage rolls on all sides.
When all have been browned, return all cabbage rolls to pan and add beef broth so that the liquid almost covers the rolls.
Simmer gently for about 1 hour.
Carefully remove cabbage rolls onto a platter, removing the toothpicks. Keep rolls warm while you make the gravy/sauce.
Mix corn starch with a bit of cold water. Add as much as needed to the liquid in the saucepan until it is the desired consistency. Season with salt and pepper if needed.
Serve cabbage rolls with potatoes (boiled or mashed) and a green or cucumber salad on the side. Pour sauce over cabbage rolls and potatoes.
