Score all around each chestnut with the tip of a knife. Bring a saucepan of water to the boil, add 1 tablespoon of the oil and the chestnuts and boil for 10 minutes. (The oil helps to loosen the skins so the chestnuts will be easier to peel.)
Remove the pan from the heat and lift out just a few chestnuts at a time with a slotted spoon. Remove the outer shell and inner brown skin – they peel much more easily when piping hot, but you will need to wear rubber gloves for this operation.
Heat the remaining oil in a large frying pan, add the carrots and chestnuts and cook, stirring, for 5 minutes, or until the chestnuts are golden brown.
Sprinkle with 100ml of water and add the thyme sprigs and fennel seeds. Cook gently for 30 minutes, uncovered, shaking the pan occasionally. Add more water if needed, then drain if necessary when the vegetables are cooked. Season with salt and pepper.
Remove the cooked thyme from the pan, tip into a serving dish and garnish with fresh sprigs to serve.