Core the cabbage and lower it into a large pot of simmering water. Remove leaves one at a time as they soften, then cool and trim the thick center ribs.
Boil the rice in plenty of salted water for 6 minutes, drain, and cool; it should remain firm.
Make the Filling and Roll:
Cook the onion and carrot in oil for 7 minutes. Add garlic for 1 minute, then cool.
Mix the pork, beef, rice, cooked vegetables, dill, salt, and pepper. Place about 1/3 cup filling near the base of each leaf, fold in the sides, and roll snugly.
Bake:
Mix the crushed tomatoes, stock, tomato paste, and paprika. Spread 1 cup in the baking dish, arrange the rolls seam-side down, and pour over the remaining sauce.
Cover tightly and bake at 350 F (175 C) for 75 to 90 minutes until the cabbage is tender and the filling reaches 165 F (74 C). Rest for 15 minutes before serving with sour cream and dill.
Notes
Use smaller inner leaves to patch any tears in larger cabbage leaves.
Freeze cooled holubtsi in their sauce for up to two months.