Preheat the oven to 350 degrees F. Butter a 9- by 13- by 1-inch glass or ceramic baking dish. Set aside.
For the vegetables: In a large nonstick skillet, heat the oil over high heat. Add the mushrooms, onions, salt and pepper. Cook, stirring frequently, until the onions are soft and any liquid from the mushrooms has evaporated, about 20 minutes.
For the sauce: In a heavy-bottomed 5-quart saucepan, bring the cream, milk and flour to a simmer over medium heat, whisking constantly for 3 minutes. Reduce the heat to low. Add the Pecorino and the mozzarella. Whisk until the cheeses have melted and the sauce is smooth. Remove the pan from the heat and stir in the spinach, basil, garlic, salt and pepper.
Add the mushroom mixture to the sauce and stir to combine. Set aside to cool slightly.
Cut the Fresh Pasta dough in quarters and press flat. Run each piece of pasta dough several times through a pasta-rolling machine adjusting the setting each time until the pasta is about 1/16-inch thick. Cut the pasta into six 12- by 4-inch rectangles (reserve any pasta trimmings for another use). The pasta dough can also be rolled out by hand. Cut the chilled dough into quarters and using a rolling pin, roll out the dough until 1/8 to 1/16-inch thick.
Spread 1 cup of the sauce over the bottom of the prepared baking dish. Arrange 2 sheets of pasta on top in a single layer. Spread 2 cups of the sauce over the pasta sheets. Repeat the layers with the remaining pasta sheets and sauce, ending with the sauce. Sprinkle with mozzarella. Drizzle with olive oil and bake until the filling is bubbling and the top is golden, 30 to 35 minutes. Cool for 20 minutes. Cut into squares and serve.
Place the cake flour, all-purpose flour, egg yolks and salt in a food processor. Pulse to combine. With the machine running, gradually add the oil then 1/3 to 1/2 cup water until the mixture forms a dough (the dough should stick together if pinched between your fingers). If necessary, add additional water, 1 teaspoon at a time if the dough is too dry.
Place the dough on a lightly floured surface. Gather the dough into a ball and knead until the dough is smooth, 5 to 8 minutes. Cover with plastic wrap and refrigerate for 30 minutes.
The lasagna can also be made using 9 dried lasagna noodles. Cook in boiling salted water for 6 to 8 minutes until just tender. Drain and cool slightly before using.