The great thing about this pasta dish is that you can easily take one of the components and serve it on its own: the pasta for another pasta dish, the mushroom ragù as a side to steak. Together in the ravioli dish, though, the flavors meld together beautifully and prove that homemade ravioli is worth the effort.
To make the pasta dough, sift together a pinch of salt with the flour onto a dry work surface. Form a well in the center of the flour. Beat together the eggs and olive oil, then add to the center of the well. Using your pointer finger, begin to stir the flour into the egg mixture, starting from the center of the well and working your way out to the edge. When the flour and egg become a cohesive mass, remove from the work surface, scraping any bits together as you go, and lightly flour the surface again. Knead the dough for 5-10 minutes until it's smooth and slightly sticky. Wrap in plastic wrap and let rest for 30 minutes.
While the dough is resting, make your mushroom ragù. In a medium-sized skillet, heat the olive oil over medium-high heat and sauté the mushrooms until they have released all of their juices and are lightly browned. Add the garlic and onion, and cook until fragrant and the onions are translucent. Next, add the red wine and simmer until ¾ of it is dissolved. Set the ragout aside until ready to serve.
Combine the ricotta, chard, crème fraîche, the 2 teaspoons of garlic, basil, salt, and pepper in a small bowl and stir to combine. Next, roll out your pasta dough to the thinnest setting on the pasta roller. Cut twenty-four 3-inch circles out of the dough and place a heaping tablespoon of the ricotta filling in the center of each. Cut 24 slightly larger circles out of the rest of the dough for the tops. Paint the edges of the bottom circles with water and press the tops together, crimping the edges with a fork to seal. Cover and refrigerate until ready to boil.
Bring a large pot of salted water to a boil and add the ravioli. Boil slowly until tender, about 5-6 minutes. When the ravioli is cooked, drain well and toss with olive oil. Serve 3 to a plate topped with a spoonful of the ragù.