In a large heavy skillet or Dutch oven, heat the olive oil over medium heat and brown the ribs well on all sides. Place the ribs in your slow cooker, and add the onions and carrots to the skillet and cook for 3 to 4 minutes or until they take on color.
Add the garlic, and cook until fragrant.
Add the red wine and beef stock to the skillet, and stir well, scraping any browned bits off the bottom of the pan and cook for 3 to 4 minutes.
Pour the beef stock and wine and carrots over the ribs, then also add the tomatoes, paste, parsley, basil, oregano, salt and pepper.
Turn your slow cooker on low for 6 hours. (If the sauce begins to thicken too much, add a cup of water or beef stock)
After 6 hours, use a fork to pull the meat off the bones of the ribs, break the meat up into small pieces, and discard the bones.
With an emulsion blender, puree until desired consistency. Reduce sauce by one third on medium heat, stirring frequently.
Cook the pasta until it is "al dente", then drain.
Add a scoop of sauce to the pasta and combine. Pass the cheese
Serve the pasta in individual bowls topped with additional sauce and a sprinkling of chopped parsley. Pass the cheese around the table.