In a large skillet, heat 3 tablespoons of olive oil over medium heat. Then add the diced onion, grated carrot, and sliced mushrooms.
Mix the cooked rice, mashed white beans, and sautéed vegetables in a bowl. Adjust the seasoning with salt and pepper; then let cool.
Place the rolls seam-side down in a deep pot. Pour the prepared tomato sauce over the rolls until they're mostly covered. Cover the pot tightly and simmer on low heat for 30 minutes.