Cook the rice in salted water until almost done, about 1-2 minutes shorter than the usual cooking time. Drain and set aside.
Melt the butter in a large frying pan over medium-high heat. Add the onions and sauté for 4-5 minutes. Reduce the heat to low and continue to cook, stirring occasionally, for another 5 minutes, or until the onions are translucent. Remove from the heat and let cool.
Once the rice and onions have cooled, add them to a large bowl along with the ground beef, marjoram, dill, paprika, salt, and pepper. Mix well and set aside.
PREPARE THE CABBAGE: Using a paring knife, carefully cut out the core of the cabbage, about 1 1/2–2 inches deep. Make sure to cut wide enough so the leaves are detached from the core, keeping any loose leaves aside.
Bring a large pot of salted water to a boil (enough water to cover the cabbage), and carefully add the cored cabbage head to the pot, core-side down, cover with the lid, and simmer for 10 minutes. Then, turn it over (it should be core-side up now) and simmer for an additional 5 minutes, or until the leaves are softened enough to be removed from the head. The time you simmer depends on the size. The leaves will turn bright green.
Carefully remove the head of cabbage and drain, saving the water for later. Let the cabbage sit for 5-7 minutes, or until the leaves are cool enough to handle. Gently pull off the outer leaves. You will need 12–16 perfect medium to large leaves for rolling the stuffing (set aside any soggy or torn leaves to line the cooking pot). Repeat for all leaves and set them aside to cool. Flatten each leaf and use a paring knife to trim out the thick center rib, making it easier to roll.
ASSEMBLE THE CABBAGE ROLLS: Take one cabbage leaf and lay it flat on the surface. Place about 1/3 cup of the filling in the center, near the bottom of the leaf (adjust the portion size based on the size of the leaf—use more for larger leaves, less for smaller ones). Fold in the sides toward the center and roll up from the end facing you, so the stuffing is enclosed (like rolling a burrito). Try to roll the leaves tightly. Repeat with the remaining meat and cabbage leaves.
PREPARE TO COOK: Line the bottom of a large, deep pot with any unused cabbage leaves. Arrange the cabbage rolls in the pot, stacking them in two layers, and place them seam side down to secure and seal the rolls during cooking. Make sure they are packed tightly together.
Stir the tomato paste into the 2 cups of chicken broth and 1 cup of water, and pour the mixture over the cabbage rolls; there should be enough mixture to cover the rolls. Bring to a boil over high heat, then cover with a lid and simmer for 1 to 1½ hours, or until the cabbage leaves are tender and the filling is fully cooked (to check, remove one cabbage roll and cut it in half to ensure it's done).
Transfer the cabbage rolls to a plate and cover, reserving the broth. While you cook the tomato sauce, the rolls will stay warm for up to 30 minutes. Strain the broth through a fine-mesh strainer.