Serves 6 for a main course, 15 as an appetizer.
To make the pasta dough by hand, put the flour in a bowl and make a well in the center. Pour the eggs in the center and use a fork to whisk the flour into the eggs, working progressively toward the outer rim of the bowl to gradually incorporate the flour into a dough. When you have a rough dough, begin kneading it with your hands. Once the dough is fairly smooth, dust the counter with flour and lift the dough from the bowl, discarding all the raggedy bits of flour. Knead the dough until its satiny smooth and firm, 10 to 12 minutes. Dust the dough with flour, wrap it in plastic, and let it rest for 30 minutes at room temperature.
To make the pasta dough in a food processor, put the flour and eggs in the food processor bowl fitted with the plastic dough blade. Attach the cover and turn on the machine until a dough forms, about 1 minute. If there is stray flour, add 1 teaspoon water at a time through the feed tube to make a cohesive dough slightly tacky to the touch. Knead the dough by hand on a lightly floured counter until it is satiny smooth. Dust the dough with flour, wrap it in plastic, and let it rest for 30 minutes at room temperature.
To make the filling, puree the garlic, onion, carrot, celery, egg, and Parmesan in a food processor, scraping down the sides once or twice, until very finely chopped. Add the ground beef, salt, and pepper and pulse 8 to 10 times until it is minced and well blended. (You can make the filling up to 1 day in advance and store it in the refrigerator until ready to use.)
Fill a small bowl with water and have a pastry brush on hand. Sprinkle a baking sheet with flour and set it aside. Dust the counter lightly with all-purpose flour. Cut the dough into 4 pieces, keeping the unused dough covered. Roll the dough through a pasta machine into wide strips accordingly to the manufacturer's instructions, progressively making them thinner until your hand is a shadow behind the sheet of pasta. Alternatively, roll out the dough by hand, dusting the counter lightly as needed to make an even, ultra-thin sheet you can almost see through.
Use a round cutter about 3 inches in diameter to cut circles very lightly with flour to prevent sticking and keep them covered to prevent them from drying out while you continue to roll out and cut the remaining dough.
Place level tablespoon-sized dollops of the filling on half of the pasta circles, brush the edges lightly with water, and top with another circle of pasta. Use your fingers to push air pockets away from the mounds of filling and press to seal the edges well. Gently pat the top of each ravioli to spread the filling out and so each one looks like a hat. Arrange them in a single layer on the baking sheet. (To freeze, put the baking sheet in the freezer until the ravioli are solid. Transfer into resealable freezer bags for storing.)
To cook the ravioli, bring a large pot of salted water to a boil over medium-high heat and have a colander set into a large bowl on hand. Reduce the heat to low so that the water is simmering steadily and slip the ravioli into it. Simmer gently until the ravioli float to the surface, 3 1/2 to 4 minutes. Collect them with a slotted spoon and let them drain in the colander briefly. Reserve 1/4 cup of the pasta water.
Meanwhile, make the sauce by melting 6 tablespoons of the butter in a large skillet over low heat. Add the sage and cook until fragrant, about 2 minutes. Ladle 1/4 cup of pasta water into the pan, bring it to a simmer, and whisk in the remaining 6 tablespoons of butter to make a light sauce. Slip the drained ravioli into the pan and shake it to coat them with the sauce before serving with black pepper and grated Parmesan to taste.
Source: Curry, Lynne. "Beef Ravioli with Sage Butter Sauce." Pure Beef: An Essential Guide to Artisan Meat with Recipes for Every Cut. Pennsylvania: Running Press Book Publishers, 2012. 96-97. Print.