Start by trimming the core of the cabbage. This will help the leaves separate easily. Place the whole head in a large pot filled with water. Cover and bring to a boil. Allow it to simmer for about 15 to 20 minutes until the leaves soften. Keep an eye on the pot, ensuring the water remains at a gentle boil.
Once the leaves are softened, drain the cabbage and let it cool. In the meantime, you can prepare the rice. In a separate pot, combine 1 cup of white rice, 1 and 1/2 cups of water, and 1/4 teaspoon of salt. Bring it to a boil, then reduce the heat, cover, and simmer for 6 to 8 minutes until all the water is absorbed. After cooking, remove it from heat and let it stand, covered, for 10 minutes.
While the rice is cooking, heat 3 tablespoons of olive oil in a large skillet over medium heat. Add the diced onion, grated carrot, and sliced mushrooms. Season with 1/2 teaspoon each of salt, smoked paprika, fennel, and black pepper. Cook for about 12 to 15 minutes, stirring occasionally. You want the vegetables to soften and for the liquid to mostly evaporate.
Next, stir in the minced garlic and soy sauce, cooking for another minute until fragrant. Once cooked, transfer the mixture to a large bowl where you'll combine it with the rice and mashed white beans. Adjust the seasoning with additional salt and pepper if needed. Allow this filling to cool slightly.
In a saucepan, heat 1 tablespoon of olive oil over medium heat. Add the tomato paste and cook it for 2 minutes, stirring constantly. Then pour in the vegetable broth, coconut milk, and granulated sugar. Bring this mixture to a boil, then reduce the heat and let it simmer until the sauce thickens. Don’t forget to taste and adjust the seasoning as needed.
Now it’s time to roll! Take a cooled cabbage leaf and trim off any thick vein to make it easier to roll. Place a generous spoonful of the filling near the base of the leaf. Roll it tightly, and fold in the sides to enclose the filling completely. Repeat this process until all leaves and filling are used.
In a deep pot, arrange the rolls seam-side down. Pour the prepared tomato sauce over the rolls until they are mostly covered. Make sure every roll has some sauce; it helps keep them moist while cooking. Cover the pot tightly and let it simmer on low heat for about 30 minutes. This allows the flavors to meld beautifully.
After 30 minutes, gently lift the rolls from the pot. They can be quite delicate, so take your time! Serve them garnished with fresh dill, and add a dollop of sour cream on the side if desired. Enjoy your homemade Vegetarian Cabbage Rolls!