Add the canned whole plum tomatoes with their juices to a blender or food processor and pulse until mostly smooth. Set aside.
Pat the beef short ribs and country-style pork ribs dry with paper towels. Season generously on all sides with kosher salt and freshly ground black pepper.
Heat the olive oil and butter in a large heavy pot over medium-high heat until the butter foams. Add about half of the meat in a single layer and brown on all sides, about 8–10 minutes total. Transfer to a plate and repeat with the remaining meat.
Turn off the heat. Carefully spoon off and discard most of the rendered fat, leaving about 3 Tablespoons in the pot.
Turn the heat back to medium-low. Add the smashed garlic cloves and cook for 30 seconds to 1 minute, just until fragrant. Do not let the garlic burn.
Stir in the tomato paste and cook for 1–2 minutes, stirring, until it softens and coats the bottom of the pot. Pour in the red wine and use a wooden spoon to scrape up any browned bits from the bottom of the pan. Let the wine simmer for 2–3 minutes to cook down slightly.
Pour in the pureed tomatoes and stir to combine. Add a pinch of kosher salt. Bring the sauce to a gentle simmer over medium heat, then lower the heat and let it simmer uncovered for 20–30 minutes, stirring occasionally, until slightly thickened and the flavors begin to come together.
Taste the sauce and adjust seasoning with additional kosher salt if needed.
Nestle the browned beef and pork ribs back into the pot along with any juices from the plate. The meat should be mostly covered by the sauce. If needed, add a little water so the meat is mostly submerged.
Bring the pot back to a gentle simmer over medium heat, then lower the heat to low. Cover the pot, leaving the lid slightly ajar, and let it simmer very gently for 6–8 hours. Stir every 30–45 minutes, turning the meat and scraping the bottom of the pot, and taste occasionally, seasoning with kosher salt as needed. If the sauce becomes too thick or the meat is no longer mostly covered, add a splash of water and keep the heat low and steady.
When the meat is very tender and pulls easily from the bone, use tongs to carefully lift all the meat out of the pot and arrange it on a serving platter. If there is a noticeable layer of fat on top of the sauce, skim off a little with a spoon.
Turn off the heat and stir in the grated Pecorino Romano or Parmesan cheese until melted into the sauce. Taste and adjust seasoning if needed.
To serve, spoon a little sauce over the meat. Cook pasta in well-salted water until al dente, then drain. Ladle some of the Sunday sauce over the hot pasta and toss until lightly coated. Finish with extra grated cheese and serve with the meat and additional sauce at the table.