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Mix the 2 cups flour, butter, eggs, and salt in a large bowl. Work with your hands to form dough.
Dust a work surface with the remaining flour and place the dough on top. Knead dough for about 10 minutes, incorporating a little extra flour as needed until it is smooth and pliable. To test if the dough is dry enough, poke it with a finger. If your finger is dry, you are ready to proceed. If your finger is sticky, add more flour, a bit at a time, and continue to knead the dough. When ready, cover with plastic wrap, and refrigerate for 30 minutes.
Combine the caciocavallo, egg, and ricotta in a medium bowl. Mixing very slowly with a wooden spoon, gently incorporate the ingredients.
Set rollers of a pasta machine at the widest opening. Take an egg-size piece of dough while keeping the remaining dough covered with plastic wrap or a clean kitchen towel. Flatten the piece of dough, dust it with flour, and fold it in half. Run this piece of dough through the pasta machine. Fold the dough in half and continue this process approximately 5 times, setting the rollers of the pasta machine to a narrower setting each time, until you eventually use the narrowest setting. The pasta sheets are ready when they are 1/16- inch wide. Continue until all the dough has passed through the machine and is the same thickness. Cover the sheets of dough with a kitchen towel and work with one sheet at a time to fill. Dab the top of the sheet you are about to fill with a wet kitchen towel or with a water-egg yolk wash. This helps create a sticky texture to help close the ravioli after you fill them. Using a 2-inch round cutter, cut out as many circles as you can get from the sheet. Set half of the circles aside, covering them with a kitchen towel so that they don't dry out. Place 1 tablespoon of filling in the center of each of the remaining circles. Top with the reserved circles. Carefully press the edges of the top and bottom circles together. To seal the ravioli, firmly press down the edges with the tines of a fork held horizontally. To prevent the ravioli from bursting when boiling, carefully poke a tiny hole in the center of each, taking care not to pierce the dough entirely. Arrange the ravioli in a single layer on a tray lined with a kitchen towel. Cook within the hour, or refrigerate up to six hours.
Bring 5 quarts of water to boil in an 8 quart pot over high heat. Add 2 tablespoons of salt. Add the ravioli and cook until they rise to the surface and are tender around the edges, about 3 to 5 minutes. Drain, reserving a cup of the pasta water.
Melt the butter in a large saucepan over low heat with 1/2 cup pasta water and the sage, being careful not to let the mixture boil. Add the drained ravioli and the caciocavallo. Toss gently to coat the ravioli with the sauce. Taste and adjust salt if needed. Serve immediately.