Season the pork shoulder all over with salt and pepper. In a large Dutch oven, heat 2 tablespoons of oil over medium-high. Working in batches if necessary, cook the pork, turning occasionally, until browned on all sides, 8 to 10 minutes total, adding more oil if the pan looks dry. Transfer pieces to a bowl as they finish. Add the remaining 2 tablespoons oil to the pan and cook the sausage, turning occasionally, until browned, 4 to 6 minutes.

Reduce the heat to medium-low. Add the remaining 2 tablespoons oil, if needed, and the onion and garlic. Season with salt and pepper, and cook until softened, 2 to 4 minutes. Add the red wine and cook, scraping up any browned bits on the bottom of the pan, until the wine is reduced by half, about 2 minutes.

Add the tomatoes and basil, then fill one of the 28-ounce cans with water. (You'll use this to rinse out the tomato can.) Add the pork shoulder and sausage to the pot, along with any accumulated juices. Bring to a simmer, then reduce heat to low. Partially cover the pot, bring to a simmer over medium-high heat, then reduce heat to a gentle simmer and cook, stirring occasionally, until the pork shoulder falls apart when shredded with a fork, 2 to 2 ½ hours.

When you're ready to eat, bring a large pot of salted water to a boil. Add the pasta to the boiling water and cook according to package instructions until al dente. While the pasta cooks, slice the sausage and shred the pork shoulder. Transfer to a platter, along with the meatballs and a few spoonfuls of sauce. Reserve ½ cup pasta water, then drain and add the pasta to the pot of sauce. Gently toss the pasta with the sauce, adding pasta water as needed to the sauce until the sauce clings to the pasta.