To make the fresh pasta, place the flour on a work surface and make a well in the center using a cup or something similar. Have a bench scraper handy to help keep the flour well intact. Combine eggs, oil and salt; pour into the center of flour well. Using a fork, begin whisking the eggs, gradually stirring in flour from bottom of well. Push some flour from sides of well into the center as the mixture begins to thicken. Continue whisking until mixture becomes very stiff.
Start to knead dough by hand and gradually work in additional flour along with 4 to 6 tablespoons of water (1 tablespoon at a time). Continue kneading and dusting with flour until dough is no longer sticky and springs back when pressed in the center. Cover ball of dough with plastic wrap and allow to rest for 1 hour. You can also use a food processor to make the dough or use store-bought.
To make the ravioli filling, place all of the ingredients in a bowl and stir together until thoroughly combined. Check for seasoning and adjust, if necessary.
When dough has rested, divide into thirds. Working with one piece at a time (cover remaining dough), run dough through a pasta machine until you achieve the desired thickness and necessary length to cover ravioli mold twice. Cut the pasta sheet in half and flour one side only of one half. Lay floured side of dough down over the ravioli mold, covering mold entirely. Place filling in the holes. Cover over with the other half of dough that has been slightly moistened with water. Press top firmly using first the palm of your hand and then passing over with a rolling pin. Place a cutting board on top of mold and turn over to release raviolis. Use a pastry wheel to cut and form individual raviolis. If necessary, lightly dust with flour and refrigerate. Repeat process until all raviolis are formed. You can also free-form the raviolis or use wonton wrappers.
Bring a large pot of water to a boil. In the meantime, to make the brown butter sauce, for each serving of about 12 raviolis, melt 3 tablespoons of butter over medium heat. When butter is fully melted, add desired amount of chopped fresh sage (I use about 6), 1-2 tablespoons of toasted chopped walnuts and a pinch of salt. Cook until ingredients come together and butter has browned a bit. Set aside.
Salt the boiling water and gently cook the raviolis just until done. Drain thoroughly, plate and top with the warmed butter sauce. Garnish with additional parmesan cheese, if desired.