Make the Pasta Dough: On a clean surface, form the flour into a mound with a well in the center. Add the eggs, salt, and olive oil into the well. Using a fork, gradually mix the flour into the egg mixture until a dough starts to form. Knead the dough for about 5-7 minutes until smooth and elastic. Wrap in plastic and let rest for 30 minutes.
Prepare the Filling: In a bowl, combine the pumpkin puree, nutmeg, cinnamon, salt, and pepper. Mix well and set aside.
Roll Out the Dough: After resting, divide the dough into quarters. Roll out each section using a pasta machine or a rolling pin, until it’s thin (about 1/16 inch thick). Make sure to dust with flour to prevent sticking.
Form the Ravioli: Place small spoonfuls of the pumpkin mixture (about 1 teaspoon) about 2 inches apart on one sheet of the rolled-out pasta. Moisten the edges around each filling with a little water. Carefully place another sheet of dough over the filling. Press down around the filling to seal, ensuring to remove any air bubbles. Cut between the ravioli using a knife or pasta cutter.
Cook the Ravioli: Bring a large pot of salted water to a boil. Carefully drop in the ravioli and cook for about 3-5 minutes, or until they float to the surface and are tender. Remove the ravioli with a slotted spoon and set aside.
Make the Sage Brown Butter Sauce: In a large skillet, melt the butter over medium heat. Once it starts bubbling, add the sage leaves and cook for about 2-3 minutes, or until the butter is golden brown and has a nutty aroma.
Combine and Serve: Gently add the cooked ravioli to the skillet to coat in the brown butter sauce. Toss lightly to combine.
Plating: Plate the ravioli and drizzle extra brown butter on top. Garnish with grated Parmesan cheese and chopped walnuts if desired.