Special equipment: pasta roller
For the filling:
Combine all the ingredients. Put it in a pastry bag and reserve until ready to use.
To roll the pasta:
Set the pasta roller on the widest setting. Start with half the pasta dough. Using your hands flatten the dough as much as you can to facilitate it going through the pasta roller. Run the dough through the roller dusting it twice with flour in between rolls if it feels sticky. Fold it into thirds, turn it 90 degrees and run it through the pasta machine 2 to 3 times. Continue to run the pasta through the machine reducing the setting in between each run, stop when you get to the second to last setting. Always hold the pasta on the tops of your hands-palms down! If you hold it with fingers up you will create stretch marks - and those are never good.
To assemble the ravioli:
Brush the lower half of the pasta (the part that is closest to you) lightly with water. This is the "glue" that will hold the ravioli together. Use the water sparingly, if you use too much it will slide and not stick. Pipe 1 inch balls of filling onto the pasta that has been brushed with water, leaving about 2 inches in between each filling ball. Fold the top half of the pasta over the pasta balls to meet the bottom edge. Using your two index fingers poke around each filling ball to seal the ravioli shut, make sure there are no air bubbles. Using a fluted round cutter cut out each ravioli. Transfer to a sheet tray coated with semolina or polenta. Reserve until ready to use.
To make the sauce:
Coat a large saute pan with extra virgin olive oil. Add the pancetta and garlic and bring to medium high heat. When the garlic is light brown and aromatic remove it from the pan and discard it. When the pancetta is crispy add the mushrooms, season with salt and saute until they are soft and wilted. Add the chicken stock, season with salt and cook until it has reduced by half. Add the butter, shake the pan frequently and cook until the saute thickens and looks velvety. Taste for seasoning and add salt if needed.
Bring a large pot of well salted water to a boil. Add the ravioli and cook for 3-4 minutes. Carefully remove the ravioli from the water and immediately put in the saute pan with the sauce. Bring to a boil shaking the pan frequently. Reserve about two tablespoons of grated parmigiano and sprinkle the rest into the sauce, stirring to combine.
Plate the ravioli, sprinkle with a little more grated parm and the chopped chives.
To make the pasta dough: In an electric mixer fitted with a dough hook*, combine the flour and salt. Add the eggs, 1 at a time, and continue to mix. Drizzle in 1 tablespoons of the olive oil and continue to incorporate all the flour until it forms a ball. Sprinkle some flour on work surface, knead and fold the dough until elastic and smooth, this should take about 10 minutes. Brush the surface with the remaining olive oil and wrap the dough in plastic wrap; let rest for about 30 minutes to allow the gluten to relax.
*Alternatively if you don't have an electric mixer: Combine the flour and salt on a flat work surface; shape into a mound and make a well in the center. Add the eggs and 1 tablespoon of the olive oil to the well and lightly beat with a fork. Gradually draw in the flour from the inside wall of the well in a circular motion. Use 1 hand for mixing and the other to protect the outer wall. Continue to incorporate all the flour until it forms a ball. Continue as directed above.
Cut the ball of dough in 1/2, cover and reserve the piece you are not immediately using to prevent it from drying out. Dust the counter and dough with a little flour. Press the dough into a rectangle and roll it through a pasta machine, 2 or 3 times, at widest setting. Pull and stretch the sheet of dough with the palm of your hand as it emerges from the rollers. Reduce the setting and crank the dough through again, 2 or 3 times. Continue tightening until the machine is at the narrowest setting; the dough should be paper-thin, about 1/8-inch thick (you should be able to see your hand through it.). Dust the sheets of dough with flour as needed.
Beat 1 egg with 1 tablespoon of water to make an egg wash. Dust the counter and sheet of dough with flour, lay out the long sheet of pasta, and brush the top surface with the egg wash, which acts as a glue. Drop tablespoons of your favorite filling on 1/2 of the pasta sheet, about 2-inches apart. Fold the other 1/2 over the filling like a blanket. With an espresso cup or fingers, gently press out air pockets around each mound of filling. Use a sharp knife to cut each pillow into squares and crimp the 4 edges with the tins of a fork to make a tight seal. Dust the ravioli and a sheet pan with cornmeal to prevent the pasta from sticking and lay them out to dry slightly while assembling the rest.
Cook the ravioli in plenty of boiling salted water for 4 minutes; they'll float to the top when ready, so be careful not to overcrowd the pot. Lift the ravioli from water with a large strainer or slotted spoon. Bath the ravioli in your favorite sauce to lightly coat and serve.