Preheat the oven to 220C/425F/Gas 7. Preheat an ovenproof frying pan.
Cream the butter and sugar together in a large bowl until light and fluffy. Beat in the eggs, one at a time, adding a little of the flour if the mixture starts to curdle. Beat in the rest of the flour with the milk and then fold in the carrots and mixed spice.
Add the olive oil to the hot frying pan and swirl it around the sides of the pan. Pour in the carrot mixture and bake in the oven for 15-20 minutes until the carrot cake is well risen and golden brown. Leave to cool slightly in the pan.
Meanwhile, to make the lime yoghurt, place the yoghurt in a small dish and stir in the lime juice.
To serve, cut the carrot cake into slices and arrange on serving plates topped with spoonfuls of the lime yoghurt.
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