Using a sharp knife trim excess fat from pork shoulder and discard. Cut pork into 4 chunks, making the pieces as even as possible. Liberally salt the pork chunks on all sides. Heat olive oil in a large skillet or Dutch oven over medium high heat, and then working in batches if necessary, brown pork on all sides. This will take about 5 minutes per side and you will know a side is sufficiently browned when it easily releases from the pan. Move browned pork to the slow cooker.
Heat the skillet/Dutch oven to medium, stir together the garlic and the anchovies (and about a tablespoon of olive oil if there isn't enough fat left in the skillet from browning the pork) until anchovies melt, garlic softens and the mixture becomes fragrant, about 1-2 minutes. Add the tomato paste and continue stirring for another 2-3minutes until the color of the tomato paste becomes a very deep ruby red and all the brown bits from the pork have released from the pan. Turn off the heat and add the olives, capers, vinegar, red pepper flakes, oregano and freshly ground pepper. Slather this sauce over each of the pork pieces.
Cover the slow cooker and cook on low for 6-7 hours. The sauce will become a rich red color and the meat will be fork tender. Remove meat to a cutting board and using two forks shred meat. Return to slow cooker, add tomatoes and heat through, about 10 -15 minutes more.
Just before serving squeeze lemon over ragu for some added brightness. I like to serve on Creamy Polenta, garnish with fresh parsley and plenty of grated Parmigiano-Reggiano