Pulse the white mushrooms in a food processor in two separate batches until finely chopped, about 15 pulses per batch. Heat the oil in a large nonstick skillet over medium heat until it shimmers, then add the shallots and garlic and cook until fragrant. Stir in the chopped white mushrooms, porcini mushrooms, and salt, then cover the pan and reduce heat to medium-low; cook for about 10 minutes until the mushrooms are wilted and wet.

Uncover the skillet and increase the heat to high. Cook, stirring occasionally, until the mushroom liquid has completely evaporated and the mixture looks clumpy and starts to brown, about 10 minutes. Stir in the cream and cook for about 1 minute until the mixture is sticky and cohesive but not wet. Transfer the filling to a bowl, stir in the Parmesan and parsley, and season with pepper. Chill in the refrigerator for 15 to 25 minutes until no longer warm.

Dust two baking sheets liberally with flour. Divide the pasta dough into five pieces and keep them covered with plastic. Roll one piece through a pasta machine to your desired thickness, then cut the sheet into 4-inch wide rectangles. Place teaspoon-sized dollops of filling along 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 firmly to seal, then use a fluted pastry wheel to cut the ravioli apart.

Transfer the finished ravioli to the prepared baking sheets and cover with a damp towel while you repeat the process with the remaining dough and filling.

Before cooking the pasta, prepare the sauce by melting the butter in a small skillet with the sage and salt over medium-high heat. Swirl the pan constantly for about 3 minutes until the butter is melted, turns golden brown, and smells nutty. Remove from the heat and stir in the truffle oil.

Bring a large pot of salted water to a boil. Cook half of the ravioli at a time, lowering the heat to maintain a gentle boil so they do not burst. Cook until tender, about 2 minutes for fresh or 4 minutes if frozen. Use a slotted spoon to transfer the ravioli to a warm platter, drizzling with some of the brown butter sauce as you go. Repeat with the second batch, then whisk two tablespoons of the pasta cooking water into the remaining sauce before pouring it over the top. Serve immediately with shaved Parmesan.