1. Bring a medium saucepan of water to a boil. Stir in the brown and wild rices, cover and cook over low heat, stirring occasionally, until tender, about 45 minutes. Drain and transfer to a large bowl.
2. Meanwhile, bring a large saucepan of water to a boil, add the cabbage, cored side down, and cook over moderately high heat until the outer leaves begin to loosen, about 10 minutes. Using tongs, carefully peel 8 outer leaves from the head of cabbage as they become loose. Reserve the remaining cabbage for another use. Cut each of the 8 leaves in half and discard the thick center rib.
3.In a medium bowl, blend the yogurt and garlic; season with salt and pepper.
4. Melt the butter in a large skillet. Add the mushrooms and lemon juice; season with salt. Cook over moderate heat until the mushrooms are tender and their liquid has evaporated, about 6 minutes. Add the shallots and cook, stirring, until softened, about 4 minutes. Add the mushrooms to the rice. Stir in 1 tablespoon of the dill and 1/2 cup of the yogurt sauce; season with salt and pepper.
5. Add the tomatoes to the skillet and cook over moderately high heat until their liquid has evaporated, about 5 minutes. Stir in the tomato paste and season with salt and pepper.
6. Preheat the oven to 375°. Spread a cabbage leaf on a work surface with the cut side facing you. Place 1/4 cup of the rice filling within 1/2 inch of the edge. Roll the leaf over the filling, tucking in the sides as you roll, to form a neat package. Repeat with the remaining leaves and filling.
7. Spoon half of the tomato sauce into a shallow 10-by-14-inch glass baking dish. Add 1/4 cup of water, arrange the rolls in the baking dish, seam side down. Spoon the remaining tomato sauce over the rolls and cover with foil.
8. Bake the cabbage rolls for 45 minutes, or until the sauce is bubbling and the rolls are hot throughout. Transfer the cabbage rolls to plates. Spoon the remaining 1 1/2 cups of yogurt sauce over the rolls. Sprinkle with the walnuts and the remaining 1 tablespoon of dill and serve.
Make Ahead: The recipe can be prepared through Step 7, covered with plastic wrap rather than foil and refrigerated for up to 3 days. Bring to room temperature before baking.
Wine Recommendation: A full-flavored rosé will complement both the earthiness of the rice and the tang of the yogurt sauce. Good options are the 2000 Château Routas Coteau Varois Rosé Rouvière and the 2000 Iron Horse Vineyards Rosato di Sangiovese from California.