Simmer barley in its 2 cups water for 30 to 35 minutes until tender, then drain and spread on a plate to cool. Heat the oven to 350°F.
While the barley cooks, core cabbage carefully with a paring knife. Lower the head into a large pot of boiling water and loosen outer leaves with tongs as they soften; remove 12 broad flexible leaves over about 10 to 15 minutes and cool them.
Shave the thick rib of each leaf level with its surface without cutting through. Mix raw beef, cooked barley, onion, garlic, paprika, dill, 1/2 teaspoon salt and pepper; divide into 12 portions.
Place a filling portion near the stem end of each leaf, fold the bottom over, tuck in the sides and roll loosely to enclose. Mix tomatoes, broth and remaining salt, spread 1 cup in a 9-by-13-inch dish and arrange rolls seam-side down in one layer.
Pour remaining sauce over the rolls, cover tightly and bake for 80 to 90 minutes until cabbage is tender and the center of each roll reaches at least 160°F. Rest for 10 minutes and serve two rolls per person.