Step 1In a food processor, mix the flour and 2 eggs until it just comes together to form a rough dough. If that doesn't happen, add a little more beaten egg, a teaspoon or so at a time. Remove the dough to a floured work surface and knead until it is smooth and elastic, about 5 minutes. Wrap with plastic wrap and refrigerate at least 30 minutes to relax the gluten.
Step 2In a mixing bowl, beat together the fresh goat cheese, Parmigiano-Reggiano, egg yolk, chives and parsley. Taste and add more salt if necessary.
Step 3Divide the dough into 4 equal parts. Work with one one piece at a time, keeping the remainder covered in plastic. Flatten the dough with your hand and then pass it through the widest setting of a pasta machine, folding the dough into thirds after the first time, and repeating until you make a thick sheet approximately 5 inches wide. Continue putting the dough through each setting, finishing with the second-to-thinnest setting; the sheet should be 6 1/2 to 7 inches wide. Lightly dust the machine or dough with flour if it sticks. The pasta will be very thin and almost translucent. Lay the pasta sheet on a floured surface.
Step 4To fill the pasta, place a small (2-teaspoon) mound of filling at 3-inch intervals, 1 inch from the edge, along the length of the pasta. Use a pastry brush or your finger to paint lightly around each mound with beaten egg white.
Step 5Fold the pasta over the filling lengthwise, making sure there's at least one-half inch from the edge of the fold to the beginning of the filling. Press down along the back edge with the side of your thumb to seal. Press between each mound of filling to seal the sides, then press along the front to make the final seal.
Step 6With a ravioli cutter, cut around the pasta to form a decorative edge, then cut between the filling, always front to back, to form the ravioli. Place the pasta on a baking sheet lined with a cotton dish towel sprinkled with flour. Repeat with the remaining pasta. If you make these early in the day, turn them over from time to time so they dry evenly.