Chill and grind: Freeze the cubed 500 grams boneless lamb leg, trimmed of silverskin and cubed on a tray until firm at the edges but not frozen through, then grind it coarse. Chilled meat grinds cleanly instead of smearing.
Brown the lamb: Heat 3 tablespoons olive oil in a wide heavy pan over medium-high heat. Add the lamb in a flat layer, working in batches if the pan is crowded, and leave it undisturbed until it browns deeply before breaking it up. Season with part of 1 teaspoons kosher salt, plus more to taste.
Soffritto: Lower the heat to medium. Add 1 medium onion, finely diced, 1 medium carrot, finely diced, and 1 celery stalk, finely diced, and cook until soft and starting to color. Stir in 3 garlic cloves, minced, 1 teaspoons fresh rosemary, finely chopped, and 0.5 teaspoons red chili flakes for 1 minute.
Paste and wine: Stir in 1 tablespoons tomato paste and cook until it darkens slightly. Add 120 milliliters dry white wine and scrape up the fond, then let it reduce by about half.
Simmer: Add 400 grams canned whole tomatoes, crushed by hand and bring to a bare simmer. Cover partially and cook, stirring now and then, until the sauce is thick and glossy and the lamb is tender. Loosen with a splash of water if it tightens too early. Adjust the salt at the end.
Cook pasta: In the last stretch of the simmer, boil 400 grams orecchiette or pappardelle in well-salted water until 1 minute short of al dente. Save about 120 ml of pasta water before draining.
Finish: Add the pasta to the ragù with a splash of pasta water and toss over the heat until the sauce clings. Off the heat, toss in half of 40 grams pecorino romano, finely grated. Serve with the rest of the pecorino and 8 fresh mint leaves, torn on top.