Bring a large pot of lightly salted water to a boil. Place cabbage head into the pot, cover, and cook until leaves are softened enough to remove from the head, about 3 minutes. Remove cabbage from the pot and let sit until leaves are cool enough to handle, about 10 minutes.

Remove 18 whole leaves from the cabbage head, cutting out any thick tough center ribs. Set whole leaves aside. Chop the remaining cabbage and spread it in the bottom of a casserole dish.

Melt butter in a large skillet over medium-high heat. Add onion and sauté until tender, 5 to 10 minutes. Remove from the heat and let cool, about 5 minutes.

Combine beef, pork, cooked rice, garlic, salt, and pepper in a large bowl. Add onion and mix until well combined.

Preheat the oven to 350 degrees F (175 degrees C).

Place about 1/2 cup beef mixture onto a cabbage leaf. Roll the leaf around beef mixture, tucking in the sides to create an envelope around the meat. Repeat to make remaining cabbage rolls. Place cabbage rolls in a layer over chopped cabbage in the casserole dish. Season with salt and pepper.

Whisk condensed soup, tomato juice, and ketchup together in a bowl; pour over cabbage rolls and cover the dish tightly with aluminum foil.