Pulse the white mushrooms in a food processor until finely chopped, about 15 pulses, working in two batches to ensure an even texture.

Heat the oil in a large nonstick skillet over medium heat until shimmering. Add the shallots and garlic and cook until fragrant.

Stir in the chopped white mushrooms, porcini mushrooms, and salt. Cover, reduce the heat to medium-low, and cook until the mushrooms are wet and wilted, about 10 minutes.

Remove the lid, increase the heat to high, and cook until the mushroom liquid has evaporated and the mixture is clumpy and starting to brown, about 10 minutes.

Add the cream and cook until the mixture is sticky and cohesive but not wet. Off the heat, stir in the Parmesan and parsley, then season with pepper.

Refrigerate the filling until no longer warm, about 20 minutes — this prevents the dough from softening too much during assembly.

Dust two baking sheets with flour. Divide the pasta dough into five pieces, keeping the unused pieces covered with plastic wrap.

Roll one piece of dough through a pasta machine into a thin sheet, then cut it into long rectangles measuring 4 inches across.

Place dollops of filling over the bottom half of the dough, spaced about 1 1/4 inches apart. Fold the top of the pasta over the filling and press the layers together securely around each mound to seal.

Cut the ravioli apart with a fluted pastry wheel and transfer to the floured baking sheets, keeping them covered with a damp towel while you finish the remaining dough.

Cook the butter, sage, and salt in a skillet over medium-high heat, swirling the pan constantly, until the butter has melted and turned a golden brown with a nutty aroma, about 3 minutes. Remove from the heat and stir in the truffle oil.

Boil the ravioli in two batches in a large pot of salted water until tender, about 2 minutes. Use a slotted spoon to transfer the cooked pasta to a warm platter, spooning a bit of the sauce over each batch to prevent sticking.

Swirl two tablespoons of the pasta cooking water into the remaining butter sauce, pour it over the ravioli, and serve immediately with shaved Parmesan.