Heat a saute pan over medium heat. Add 1/4 cup of the olive oil and the salt pork and saute until the pork is browned and most of fat is rendered, 5 to 8 minutes. Drain off and discard the fat, and set the crispy salt pork aside.
Add the red and yellow onions and a pinch of salt to the saute pan (use the salt sparingly as the salt pork is quite salty). Allow the pan to sit, covered, over very low heat so that the onions can clean the pan of the browned pork bits, 3 to 5 minutes. Add the remaining 1/2 cup olive oil, 1 teaspoon of the oregano and the garlic. Saute until the onions are soft, 8 to 10 minutes.
Add the sausage, ground beef and a little more salt and mash the meat with a spoon or potato masher to eliminate any large pieces. Saute until the meat loses its pink color. Add the remaining 2 teaspoons oregano and the red wine. Cook until the wine has evaporated and the alcohol has cooked off, 8 to 10 minutes. Add the tomato puree, the reserved salt pork, 2 cups of the stock and the tomato paste. Bring to simmer and cook over low heat, uncovered, for about 1 1/2 hours. Add the remaining 1 cup stock as needed if the ragu becomes too thick.
Transfer about 4 cups of the ragu to a large pan and keep warm over medium-low heat. (Freeze the remaining ragu for future use.)
Bring a large pot of water to a boil and add some salt. Cook the orecchiette until al dente.
Just before the pasta is fully cooked, add the butter and parsley to the ragu pan. Drain the pasta and add it to the ragu pan, allowing it to cook together for a couple of minutes so that the pasta will absorb all the excess liquid, giving it a rich glaze. If the pasta is too dry, add a small amount of pasta cooking water. Add all but 1 tablespoon of the Parmigiano-Reggiano and Pecorino-Romano, remove from the heat and toss to combine.
Transfer to a large serving platter and top with the remaining cheese.
This ragu is best served the day following its preparation. You should be left with about 8 cups (2 quarts) of the ragu after this recipe. Freeze for future use; it makes a great busy weeknight dinner.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.