Boil the potatoes for 20 mins, adding carrots for the last 5 mins.
Drain, spoon out the carrots, then mash the potatoes with cream and butter. Fry the carrots, turnips and garlic in oil for 10 mins; add the cabbage for the final 2 mins.
Mix with the mash in an ovenproof dish and season. Can be made up to 1 day ahead.
To serve, grill for 5-10 mins or, if made ahead, bake at 200C/180C fan/gas 6 for 20 mins.