Combine the ricotta, goat cheese, lemon zest, egg yolk, 2 tablespoons of Parmesan, extra-virgin olive oil, chives, tarragon, salt and black pepper in a small mixing bowl. Stir to blend. Cover the bowl with plastic wrap and refrigerate until ready to use.
Trim 3 to 4 pieces of parchment paper to fit a baking sheet. Use one piece of the paper to line the baking sheet, set the remaining pieces aside. Lightly dust the parchment paper-lined baking sheet with all-purpose or semolina flour.
Cut the pasta dough into 4 portions. Working with one portion at a time, (and keeping the rest loosely covered with a damp towel on top), flatten the dough into a disk with the palm of your hand. Beginning with the widest setting on a pasta machine, roll the dough through the machine, folding the pasta into thirds after each “run.” Run it through about two times at each setting, decreasing the settings until you get to the thinnest setting, at which point, you can just run the pasta through once. Place the rolled pasta dough on the prepared baking sheet. If the piece of dough is longer than the baking sheet and you need to fold it over, make sure you dust with flour to prevent sticking. Cover with one of the reserved pieces of parchment paper and a damp towel while you repeat with the remaining portions of dough, placing a piece of parchment between layers.
Form the ravioli by spacing small mounds of the cheese filling (about 1 tablespoon each) onto one of the pasta sheets in rows, leaving about 1 ½ inches of space between the mounds and around the sides for sealing. Run a wet finger (or pastry brush) around the edges of the filling mounds, making sure to wet all exposed edges. Place a second sheet of pasta over the top of the filled sheet, pressing the top sheet down and easing it around the mounds of filling so that each ravioli is sealed. Cut into individual ravioli using a round cookie cutter (about 3 ¼ inches in diameter) or a ravioli stamp, or cut into squares using a sharp knife. (Save the pasta scraps in case you need more pasta to use all the filling -- you can rerun scraps through the machine again to form additional pasta sheets if necessary). Seal each ravioli by firmly pinching the edges together and pressing out any air bubbles between the pasta sheets. Using a fork, pressing the edges together firmly to seal. Repeat until you have used up all of the filling and pasta.
Fill a large pot with salted water and bring it to a boil. Place the ravioli in the boiling water (in batches, if necessary) and cook until the ravioli come to the surface of the water and the pasta is al dente, 3 to 4 minutes. Remove the ravioli with a slotted spoon and place on a paper towel-lined plate to drain. Cover lightly with a towel to keep warm.
Combine the butter, chives, sage, and crushed red pepper in a 12-inch sauté pan and heat over medium heat, stirring, until the sauce just starts to bubble. Remove from the heat.
Divide the ravioli among serving plates and spoon the sauce over the top. Garnish with the chopped parsley and additional Parmesan, if desired. Serve immediately.