Preheat an oven. Heat 2 Tbsp olive oil in a large Dutch oven over medium-high heat. When hot add the lamb shoulder, working in batches as needed so as not to overcrowd the pan, then season with salt and freshly ground black pepper. Brown on all sides.
Use a slotted spoon to transfer the browned lamb to a large plate before sautéing the aromatics, or soffritto (onions, carrots, and celery).
Cook before adding garlic, tomato paste, and fresh herbs, then deglaze with white wine. Be sure to scrape up any browned bits on the bottom of the pan. Add crushed tomatoes to make the sauce.
Add the browned lamb, along with any juices, bring to a simmer, then cover with a tight fitting lid and transfer to a preheated oven. Cook low and slow until the lamb is tender and falling apart.
Stir in the last remaining ingredients: capers, olives, lemon zest, lemon juice, balsamic vinegar, and basil. Taste, adjust seasoning, then serve!
Toss until thoroughly coated, then serve with grated Parmigiano Reggiano and lots of fresh basil.