Combine sauce ingredients; set aside. Line a medium bowl with cheesecloth, allowing 6 inches to overhang edges of bowl. Place cabbage in bowl with core facing up. Remove core and center of cabbage, leaving a 1-in. shell. Discard core; chop center cabbage leaves.
Heat large skillet over medium heat; add oil. Cook onion and 1 cup chopped cabbage in oil until tender, 4-5 minutes. Remove from the heat and cool slightly. In a small bowl, combine beef, onion mixture, 1 cup sauce, rice, egg, salt and pepper. Spoon mixture into cabbage shell. Gather cheesecloth around cabbage and twist tightly to securely enclose cabbage leaves and filling.
Place 1 cup water, remaining chopped cabbage and remaining sauce in a Dutch oven; mix well. Carefully add stuffed cabbage, with twisted cheesecloth facing down. Bring to a boil. Reduce heat; simmer, covered, until whole cabbage is tender and a thermometer inserted in the middle of the cabbage reads 165°, 1 - 1-1/4 hours. Remove cabbage and discard cheesecloth. Place cabbage on a serving platter; keep warm.
Combine the cornstarch and remaining 1/4 cup water; add to Dutch oven. Bring to a boil, stirring constantly. Boil until thickened, about 2 minutes. Cut cabbage into wedges; serve with sauce and Parmesan cheese. If desired, sprinkle with fresh thyme and oregano leaves.