Prepare the Filling Vegetables: Chop onion, leeks, carrot, and garlic finely using a mixer or knife. Heat olive oil in a large pan and add the chopped vegetables. Sauté gently to caramelize and soften.
Prepare the Mushrooms: Chop mushrooms finely in batches using a mixer without over-processing. Add mushrooms to the pan with the vegetables.
Cook the Filling: Add white wine and two vegetable broth cubes to the pan. Cook for 10-15 minutes, allowing moisture to evaporate and flavors to concentrate. Stir occasionally.
Finish the Filling: Add grated cheese and chopped parsley to the filling mixture. Stir well and set aside to cool slightly.
Fill the Cannelloni: Using a piping bag or by hand, fill the cannelloni tubes evenly with the mushroom filling. Place filled cannelloni in an ovenproof dish.
Prepare the Tomato Sauce: In a bowl, mix tomato juice, water, granulated sugar, fresh basil, salt, and pepper. Pour the sauce evenly over the filled cannelloni.
Prepare the Yogurt Crust: In a bowl, whisk together strained yogurt, eggs, salt, pepper, and grated cheese. Spread this yogurt mixture evenly over the tomato sauce-covered cannelloni. Sprinkle extra cheese on top.
Bake the Dish: Cover the dish with aluminum foil and bake in a preheated oven at 180°C (356°F) for 30 minutes. Remove the foil, increase the temperature to 200°C (392°F), and bake for another 30 minutes until the crust is golden.
Rest and Serve: Allow the dish to cool and set by refrigerating for about 1 hour before cutting into portions and serving.