Preheat the oven to 300°F (150°C).
Heat the olive oil in a Dutch oven over medium-high heat. Sprinkle the lamb shanks with 2 teaspoons salt and sear them in the pot, turning to brown them on all sides. Remove the shanks to a plate and pour off any excess fat from the pan.
Pour the tomatoes into a large bowl and, using your hands, shred them into small pieces. Set aside.
Add the onions, fennel, carrots (if using), garlic, and the remaining 1 teaspoon salt to the pot. Cook, stirring and scraping any browned bits off the bottom of the pot, until the garlic is lightly toasted, 3 to 4 minutes. Squeeze in the lemon juice from each half and toss the lemon rinds into the pot. Stir in the wine, tomatoes, and ginger, scraping any browned bits up from the bottom of the pot, and bring to a simmer. Cook for about 3 minutes.
Stir in the cinnamon, coriander, fennel seeds, and 3/4 cup water, then add the lamb to the pot. Cover and braise in the oven until the lamb shanks are fork-tender and falls easily off the bone, about 3 hours.
Arrange the braised lamb shanks on a platter. Stir the butter into the pan juices and, once melted, add the mint. Pour the sauce over the lamb, discarding the cinnamon sticks. Sprinkle with parsley, if desired, and serve with mashed potatoes.