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.