In a large Dutch oven or saucepan, add olive oil and butter and heat. Add onions, celery and carrots and cook until very soft and beginning to caramelize. Mix together all of the meats. Add the meats to the pan and begin to brown. When the meat begins changing color and releasing its own liquids, add the milk. Cook until the milk is almost totally evaporated--it should just be moist around the edges of the meat, about 15 minutes. Add the wine. Add the tomato paste and stir well. Bring to a simmer. Reduce heat to low and cook for 2 hours.
Roll out pasta dough to the thinnest setting on a pasta machine. Cut into strips that are 4-inches wide and 8 inches long. Starting with the 4-inch side, loosely roll the pasta into a tube that is about 4-inches long and 2 1/2-inches wide. Cut the open side into 1/4-inch wide strips. Unroll the pasta and place in small bundles.
Bring 6 quarts of water to a boil. Add salt to the water and return to a boil. Add the tagliatelle and cook for 5 minutes. Drain the tagliatelle and add to the bolognese sauce. Thin with a little pasta water, if necessary. Toss for 1 minute. Immediately serve in warm pasta bowls. Top with freshly grated Parmigiano-Reggiano.
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 reflour 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 or shape as desired.