Make the cabbage: Bring a large pot of water to a boil. Remove 4 or 5 unblemished outer leaves from cabbage, and reserve. Add cabbage head and 2 tablespoons salt to pot. Cover and simmer, flipping once, until tender, about 45 minutes. Transfer cabbage to work surface, then add reserved leaves to pot, and cook until bright green and tender but not mushy, about 3 minutes. Spread in a single layer on paper towels to drain.
Make the filling: Bring a medium pot of water to a boil. Add rice and 1/4 teaspoon salt, and simmer until tender, about 15 minutes. Drain, rinse rice with cold water, and drain again. Toss with 2 tablespoons oil in a large bowl.
Place currants in a small bowl, and add enough warm water to cover. Let stand until plump, about 15 minutes.
Heat 2 tablespoons oil in a medium saucepan over medium heat. Add chard stems and 1/2 teaspoon salt, and cook, stirring occasionally, for 3 minutes. Add water, and simmer until tender, about 12 minutes. Add onion, and cook, stirring occasionally, for 5 minutes. Remove from heat.
Heat remaining 2 tablespoons oil in a large skillet over medium-high heat. Add chard leaves, and cook until barely wilted and bright green, about 2 minutes. Add spinach, and cook until wilted and bright green, about 2 minutes more. Remove from heat. Let cool for 10 minutes.
To assemble: Preheat oven to 350 degrees. Cut out stem from cooked cabbage, and discard. Using a spoon, scoop out inside of cabbage, leaving a 1/2-inch-thick shell. Squeeze out excess liquid from scooped cabbage, and coarsely chop insides, then stir into rice. Dry cabbage shell.
Stir together the spinach mixture, chard stems, currants, apricots, pine nuts, parsley, eggs, vinegar, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper into rice mixture.
Lightly oil a rimmed baking sheet. Fill cabbage shell with rice mixture. Place on a clean kitchen towel, and gather ends of towel around cabbage, squeezing gently to form a round shape and remove excess liquid. Remove towel, and flip stuffed cabbage, open side down, onto baking sheet. Rub butter over entire surface of cabbage, and bake until heated through, about 30 minutes.
Drape reserved cabbage leaves over stuffed head, overlapping, to cover. Smooth leaves down and tuck underneath cabbage. Cut into wedges, and serve immediately.