In a medium Dutch oven, warm 2 tablespoons of the oil over moderate heat. Season the pork with salt and pepper, add to the pot and brown well on all sides, about 15 minutes total. Spoon off the fat. Add the leek, halved carrot, mint, oregano and wine and bring to a simmer, scraping up any browned bits from the bottom of the pot. Cover and cook over low heat, turning the meat every 20 minutes, until very tender, about 2 hours. (The pork can be left to cool in the pan juices, then refrigerated for up to 2 days.)