To prepare the pasta dough, on a clean work surface, place the flour in a pile and use a fork to create a well in the middle with high, raised sides. Crack the eggs into the well, then add the olive oil, water and salt. Using the fork, beat the eggs to slowly incorporate the flour by pulling it in from the sides of the well. As you continue to pull in more flour and mix, the dough will start to come together.

Using your hands, the fork or a bench scraper, work the dough until it comes together. Add more flour if it is sticky; add a few splashes of water if it is too dry. Knead the dough until smooth and elastic, 8 to 10 minutes. 

Wrap the dough in plastic wrap and let rest for 20 minutes.

To prepare the ravioli filling, in a bowl, combine the duck confit, egg, ricotta, Parmesan, thyme and parsley. Fold the ingredients together until blended. Set aside.

Sprinkle a large baking sheet with flour or cornmeal. Unwrap the pasta dough and cut it into 4 equal pieces; you will work with 1 piece at a time. Wrap the remaining 3 pieces in plastic wrap so the dough won't dry out. 

Flour a work surface. Using a floured rolling pin, roll out the dough, working from the middle and rolling out. Try to get the thinnest and longest piece of dough you can to cut into 2 pieces for a frame. Keep rotating and turning the dough to prevent sticking. If using a ravioli maker, you will need 2 long rectangular shapes.

Once your dough is rolled very thin, place 1 strip of the dough over a ravioli frame and press the dough into the frame with the indented tray. Fill each indent with about 1 teaspoon of the duck confit filling; do not overfill. Place the second strip of pasta over the strip with the filling. Press the strips together with your fingers.

Gently roll a rolling pin over the frame until the frame is visible. Remove the ravioli and trim out the squares using a knife or a ravioli cutter. You may need to pinch the sides of the ravioli to make sure it is sealed. Place the ravioli on the prepared baking sheet. Remove the excess dough and reroll. Repeat the process with the remaining dough and filling. Let the ravioli dry for 20 minutes before cooking. (The ravioli may be frozen at this point for later use.)

To cook, bring a large pot of salted water to a boil. Drop the ravioli in and stir gently. Cook for about 5 minutes, then drain well. Return the ravioli to the hot pot and drizzle with a little olive oil to prevent them from sticking. Place the pot on the turned-off burner to keep the ravioli warm while you make the sauce. 

To prepare the mushroom sauce, in a large sauté pan over medium-high heat, warm the olive oil. Add the shallot and mushrooms and cook, stirring occasionally, until the mushrooms begin to brown, about 5 minutes. Add the garlic and thyme and cook, stirring, until fragrant, about 30 seconds. Add the wine, stirring to scrape up any browned bits, and bring to a boil, then simmer for about 2 minutes. Add the stock and bring to a boil, then simmer for about 2 minutes.

Reduce the heat to low, add the cream and simmer until the sauce is thickened, about 3 minutes. If it is too thick, add a bit more stock. Adjust the seasoning with salt and pepper.

Pour the sauce over the cooked ravioli, sprinkle with Parmesan and serve.