Season the lamb well with salt and freshly ground black pepper.
Heat a large flameproof casserole dish over a medium heat until hot. Add the oil and lamb and cook, turning occasionally, until golden-brown all over. Remove from the pan and set aside.
Add the onions, carrots, turnips and thyme to the same casserole and fry for 1-2 minutes.
Return the lamb to the casserole and pour over the stock. Place into the oven to cook uncovered for 1½ hours, or until the meat is completely cooked through and tender. Remove from the oven, cover and allow to rest for 15 minutes.
For the mashed potatoes, place the potatoes and mint into a pan of salted water bring to the boil, then reduce the heat and simmer for 15 minutes, or until the potatoes are tender. Drain the potatoes, return to the pan and place back over a low heat to drive off any excess moisture.
Add the butter and mash thoroughly. Season, to taste, with salt and freshly ground black pepper.
For the mint sauce, place the chopped mint into a small bowl.
Place the vinegar and sugar into a non-reactive pan and bring to a simmer. Pour the mixture over the mint and stir well to combine. Set aside until the lamb is ready.
To serve, carve the lamb and place equal amounts onto each plate. Spoon the vegetables alongside with a dollop of mash. Spoon the mint sauce over the lamb.