First, make the caramelised carrots. Put the carrots in a saucepan so they can sit in just about a single layer. Add the orange juice, butter, sugar and enough water to cover the tops of the carrots by just 1cm. Bring to the boil, then cook until the water has almost evaporated and the carrots are left in a sticky syrup – you may want to reduce the heat if the liquid looks more syrupy, and go a little slower at the end so the carrots don’t catch. Can be made up to 1 day ahead.
Heat oven to 160C/140C fan/gas 3. Grease a 23cm cake tin. Lift the candied carrots from the pan and syrup, and arrange in the base of the tin, cut-side down. Keep the pan and syrup for later.
Whisk together the 200g brown sugar, the oil, yogurt, eggs, vanilla and zest from 2 oranges. Mix the flour, mixed spice, cinnamon and grated carrot in a big mixing bowl. Stir in the whisked mixture until smooth, then spoon over the carrots in the tin – be careful not to dislodge their arrangement too much. Bake for 45-50 mins until a skewer poked in comes out clean. Cool in the tin for 20 mins.
Meanwhile, add the orange juice to the syrup pan with the 3 tbsp brown sugar. Simmer together until slightly reduced, then stir in the remaining zest.
Turn the cake out onto a plate and spoon over the syrup. Eat just warm or at room temperature with more yogurt or crème fraîche.