Heat ¼ cup oil in a large sauté pan over high heat. Add as many lamb cubes as will comfortably fit in the pan (overcrowding will cause the lamb to steam rather than brown) and brown on all sides; transfer to a deep casserole. Repeat with the remaining lamb cubes, adding 1 to 2 tablespoons oil to the pan as needed for each batch.
Heat 2 tablespoons oil in the same sauté pan over medium heat. Add the onions and cook until translucent, 7 to 10 minutes; add to the casserole with the lamb. Pour the wine into the pan and boil a minute or two, scraping up the browned bits on the bottom; add to the casserole. Pulse the tomatoes with their juices in a food processor until chopped and add to the casserole along with the garlic. Heat to boiling. Reduce the heat, cover, and simmer very slowly until the meat is tender. (You will not need any stock; just let the meat and tomatoes give off their juices to make a very rich sauce.) Add the parsley for the final 10 minutes of cooking. Season to taste with salt and pepper. Skim well. (The stew may be prepared ahead up to this point.)
Just before serving, add the greens to the stew, cover, and let them wilt in the pan juices over low heat. Sprinkle with cheese and serve immediately.