Mound the flour on a clean surface, create a well in the center, and add the whole eggs, egg yolks, olive oil, and salt. Beat with a fork, slowly incorporating flour from the inner walls until a shaggy dough forms, then bring together with your hands.
Knead the dough firmly with the heel of your hand for 8 to 10 minutes, pushing forward, folding back, and rotating a quarter turn, until smooth and elastic. If too dry, add cold water one teaspoon at a time; if sticky, dust lightly with flour.
Wrap the dough tightly in plastic wrap and let it rest at room temperature for at least 30 minutes. This relaxes the gluten and makes rolling much easier.
Wilt the fresh spinach by pouring boiling water over it in a colander, press out all excess moisture, then finely chop. (If using frozen spinach, thaw and squeeze completely dry.)
In a medium bowl, combine the ricotta, chopped spinach, Parmigiano-Reggiano, egg yolk, nutmeg, salt, and pepper. Stir until fully combined and thick; refrigerate for 15 minutes if the mixture seems too loose.
Cut the rested dough into 4 equal portions, keeping unused pieces covered. Using a pasta machine (setting 5 or 6) or rolling pin, roll each portion into a thin sheet, roughly 1-2mm thick, until you can faintly see your hand through it.
Place 1-teaspoon mounds of filling along one half of a pasta sheet, spacing them about 1.5 to 2 inches apart, then lightly dampen the pasta around each mound with water using your fingertip.
Fold the other half of the pasta sheet over the filling mounds, press firmly around each to remove air pockets, then cut into individual ravioli with a fluted pastry wheel or sharp knife. Press the edges firmly to seal and place on a floured parchment-lined baking sheet.
Bring a large pot of generously salted water to a rolling boil, then carefully lower the ravioli in batches. Cook for 3 to 4 minutes until they float to the surface and the pasta looks cooked through, then remove with a slotted spoon.
Melt the butter in a wide skillet over medium heat, add the sage leaves, and cook, swirling occasionally, until the butter turns golden amber and smells nutty, about 3 to 4 minutes. Add the cooked ravioli, toss gently to coat, and serve immediately topped with freshly grated Parmigiano-Reggiano.