Bring a pot of water to a boil and season with salt. Boil cabbage leaves to soften, a couple of minutes. Drain and cool to handle.
In a saucepot over medium heat, melt 2 tablespoons butter and add pasta. Toast to deeply golden and add rice, stir 1 minute, add 2 1/2 cups stock and cook at a simmer for 15 minutes. Cool and reserve.
Melt remaining 2 tablespoons butter in a large, deep skillet over medium heat. Add the meat and lightly brown. Season with salt and pepper; add currants, onions, garlic, nutmeg, dill and parsley. Transfer to a bowl with the rice and stir to combine. To the skillet, add tomato sauce, diced tomatoes and remaining cup stock, and stir to combine.
Fill and roll cabbage leaves to make 10-12 small burrito-shaped logs. Arrange in the skillet with the sauce and bring to a boil. Cover and cook over low heat for 30 minutes.