Bring a large pot of salted water to a boil. Place the whole cabbage head into the pot and cook for about 2 minutes. As the outer leaves soften, gently pull them away using tongs and set them aside to cool.
Preheat the oven to 350°F (175°C) so it is ready when the rolls are assembled.
Cook the rice according to package directions but stop about 5 minutes early so it remains slightly firm.
In a large skillet over medium heat, cook the ground pork or turkey, ground beef, diced onions, garlic, dill, salt, and black pepper. Stir often and break the meat into small pieces until no pink remains. Drain excess fat.
Add the partially cooked rice, diced tomatoes with their juices, and about 1/3 cup of the tomato sauce to the skillet. Stir well and allow the mixture to cool slightly.
Mix the beaten egg into the cooled filling to help bind the mixture together.
In a bowl, combine the remaining tomato sauce and the condensed tomato soup. Spread a thin layer of this sauce mixture on the bottom of a 9x13 inch baking dish.
Lay each cabbage leaf flat on a cutting board. Trim the thick center rib with a knife so the leaf folds easily.
Place about 1/4 to 1/3 cup of filling in the center of each leaf. Fold the sides inward and roll the cabbage tightly around the filling. Place each roll seam side down in the prepared baking dish.
Pour the remaining tomato sauce mixture evenly over the cabbage rolls. Cover the dish tightly with foil.
Bake for 75 to 90 minutes until the cabbage is tender and the flavors have blended.
Remove from the oven and allow the cabbage rolls to rest for about 15 minutes before serving.