Prepare the pasta dough according to the recipe and roll out to the thinnest setting on a pasta-rolling machine. Set aside.
In a Dutch oven, heat the olive oil over high heat, and add the onion. Cook for a few minutes, until it is softened and golden brown, then add the sausage and beef. Season with salt and pepper and cook until the meat begins to brown. Stir in the crushed tomatoes and their juices, as well as the hot pepper. Cook over medium heat for 15 minutes, stirring to meld the flavors and make sure the meat is cooked all the way through.
Divide the sauce into 2 equal portions and place in separate bowls. Into 1 portion, add the eggs and cheese and stir to make a homogenous mixture. Allow this portion of the mixture to cool completely. Transfer the nonegg portion of the mixture to a saute pan.
Cut the pasta sheets into 4-inch squares and place a heaping teaspoon of the mixture with egg and cheese added in the center of each square. Fold each square over on itself to form a rectangle, and seal the edges with finger pressure.
Bring 6 quarts of water to a boil in a large pot over high heat, and add 2 tablespoons salt. Cook the packets in the boiling water for 4 minutes, then drain, and add to the saute pan with the ragu. Toss over high heat for 1 minute, then serve, divided evenly among 6 warmed pasta bowls, topped with grated cheese.
Basic Pasta Dough:
Mound 3 1/2 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and the olive oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour, starting with the inner rim of the well.
As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board, and scrape up and discard any leftover bits. Lightly re-flour the board and continue kneading for 6 more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature.
Roll the pasta out on a pasta-rolling machine to the desired thickness.