Remove the mushroom stems, clean the caps with damp paper towels, slice thinly, and cook in a pan with butter, oil, and salt until the mushroom liquid is completely absorbed and the mushrooms are soft. Allow to cool before chopping the mushrooms with a knife. Set aside.
Cook the tagliatelle in boiling salted water until al dente, about two to three minutes.
Meanwhile, place the béchamel sauce in a large bowl (set aside a couple of tablespoons).
When the pasta is al dente, set aside a cup of the cooking water, drain, and pour the tagliatelle into the bowl with the béchamel sauce.
Meanwhile, preheat the oven to 190°C (375°F).
Add 1 or 2 ladles of the pasta cooking water (about 4 tablespoons), to the tagliatelle, then mix vigorously.
Now add the mushrooms and ham (or cheese), more pasta cooking water, a drizzle of olive oil, 50 g of Parmigiano cheese, and mix.
Use a large fork to form 4 nests or small mounds. Arrange the nests side by side in a baking dish, with a few flakes of butter placed in the bottom. Spread the remaining sauce from the bowl over the nests, add a scant tablespoon of béchamel, sprinkle with Parmigiano.
Cover the baking dish with aluminum foil, place in the oven, and remove the foil after 5 minutes. Continue baking for another 5 minutes or until the pasta is gratinated to your liking.