Step 1In a food processor, pulse the flour and the salt to mix. Add the whole eggs and 1 egg yolk and process until the dough begins to come together in a large ball, about 2 minutes. If the dough only gets sandy, it is too dry and you will need to add the extra egg yolk.
Step 2Gather the dough into a ball and knead it on a lightly floured surface just until smooth, 4 or 5 strokes. Wrap tightly with plastic wrap and set aside for at least 30 minutes to rest.
Step 1Heat the olive oil in a large skillet over medium heat. When hot, add the garlic and cook, stirring, until fragrant, about 1 minute. Add the chopped beet greens and cook until they are tender and wilted, about 3 minutes. Turn the cooked greens onto a chopping board and mince fine.
Step 2Place the cooked greens in a mixing bowl and stir in the ricotta, Parmigiano-Reggiano, salt and a generous grinding of black pepper. Stir together until fairly uniform. Taste and adjust seasoning for salt, Parmigiano-Reggiano and pepper. Stir in the egg yolk. Cover and refrigerate until ready to use.
Step 1Cut the pasta in quarters or eighths, depending on how proficient you are at rolling dough -- the smaller amounts are easier to manipulate. Work with one portion at a time, keeping the rest wrapped.
Step 2Press the dough ball into a flat disk and lightly flour both sides. Roll it through the pasta machine set at the widest setting. Fold the dough into thirds and repeat as often as necessary until the dough is silky, 4 or 5 passes. Flour lightly as necessary.
Step 3When the dough is silky, begin stretching it. Adjust the pasta machine a notch tighter and roll the dough through. Flour lightly, adjust the machine a notch tighter and repeat. Keep doing this until you are at the narrowest or next-to-narrowest setting. You should have a long ribbon of dough of roughly uniform width.
Step 4When the dough has been rolled thin enough, place it on a lightly floured surface and fold it over lengthwise, matching up the ends to make as neat a rectangle as possible. Trim the rounded ends. Cut the dough sheet in half at the point that it folded over and set one of the halves aside.
Step 5Give the filling a good stir and then place it in generous three-fourths-teaspoon balls on the pasta sheet, roughly 1 inch apart in 2 parallel rows. Carefully lay the reserved pasta sheet over the top, matching up the ends. Press down firmly around each ball of filling to seal the dough, squeezing out as much air as possible. Cut the pasta into individual ravioli and set them aside on a lightly floured towel. Be careful not to let them touch -- they will want to stick together. Repeat using the remaining dough and filling; you may have some filling left over.
Step 6When you're almost ready to serve, melt the butter in a small skillet with the sage. Bring a large pot of water to a boil and salt it generously. Add the ravioli to the boiling water and cook 4 to 6 minutes. To be absolutely sure the pasta is tender enough, pull one ravioli and cut a piece from the edge. It should be tender, but still have some resistance.
Step 7While the ravioli are cooking, rinse out a large bowl with a ladleful of hot pasta water and dry it with a towel. When the pasta are done, drain them from the water and empty them into the bowl. Pour over the melted butter and sage leaves, season with salt and gently stir to combine. Sprinkle grated Parmigiano over the top and bring immediately to the table.