Boil the potatoes unpeeled in salted water until soft. Drain, peel, and mash them with a bit of milk and butter until creamy yet slightly rough. Season with salt and pepper.
In a skillet, heat some olive oil over medium heat. Sauté the finely chopped shallot or onion until it begins to soften.
Add the sliced cremini mushrooms and rosemary (or parsley), and cook for a few minutes. Then add the porcini. Increase the heat, add the white wine, and allow the alcohol to evaporate. Reduce the heat and simmer until the mushrooms are tender. Set aside some cooked mushrooms for topping before baking.
In a bowl, combine the mashed potatoes, half of the remaining Parmigiano Reggiano, and the majority of the cooked mushrooms (reserve some for topping). Use a hand blender to break down larger pieces but don't overly blend to maintain some texture.
Melt the butter in a saucepan over medium heat. Sift in the flour to make a roux and cook for 1-2 minutes while mixing occasionally.
Gradually pour in the milk, stirring continuously to prevent lumps.
Stir in a third of the grated Parmigiano Reggiano and continue stirring over low heat until the sauce thickens. Season with salt and nutmeg.
Preheat the oven to 200°C (390°F).
Using a teaspoon or a piping bag, fill each cannelloni tube with the potato and mushroom mixture from one end to the other.
Spread some béchamel sauce in the bottom of a baking dish then add the mushroom cannelloni.
Cover the filled-cannelloni with the remaining béchamel sauce. Sprinkle the reserved cooked mushrooms and the rest of the grated Parmigiano Reggiano on top.
Cover the dish with aluminum foil and bake for the first 15 minutes. Remove the foil and continue baking for another 15-25 minutes until the top is golden and crispy.
Let the dish stand for 5 minutes after baking before serving.