Line a baking pan with parchment paper. Arrange the pork evenly in the pan and freeze, along with a blade, for 20 minutes.
Meanwhile, in a large pot of boiling, salted water, cook the pasta until al dente. Drain, reserving 2 cups of the pasta cooking water, and transfer to a large serving bowl.
In a large skillet, heat 2 tablespoons olive oil over high heat. Add the mushrooms and cook, stirring, until softened, about 3 minutes. Lower the heat to medium, add the garlic and scallion whites and cook until softened, about 3 minutes. Add the tomatoes and their liquid and simmer for 10 minutes.
Fit the chilled blade into the food processor. Working in 2 batches, add the pork, red pepper, 1 teaspoon salt and 1/2 teaspoon black pepper and pulse until the meat is ground. In a large skillet, heat the remaining 2 tablespoons olive oil over medium-high heat. Add the pork and cook, stirring occasionally, for 5 minutes. Transfer to the tomato sauce. Add 1 cup pasta cooking water to the skillet and scrape up the browned bits; add to the sauce.
Simmer the sauce over low heat for 5 minutes. Pour over the pasta;add the parsley, scallion greens and cheese and toss, adding more pasta water if necessary. Pass extra parmesan at the table.