For the cake:
For the cake, preheat the oven to 190C/370F/Gas 5.
Lightly grease a 26cm/10in round cake tin (that's 6½cm/2½in deep) cake tin and line the base with baking parchment.
Whisk the sugar, eggs and olive oil together in a large bowl using an electric hand whisk for 3-4 minutes until the mixture is smooth.
Sift the flour, mixed spice and bicarbonate of soda into the bowl, tipping in the grains left in the sieve. Stir in gently. Fold the grated carrot, pecan nuts, lemon zest, sultanas, raisins and desiccated coconut into the mixture.
Spoon the mixture into the cake tin and smooth the surface with a spatula. Bake on the centre shelf of the oven for 35-40 minutes. It should be nicely risen and feel firm and springy to the touch when lightly pressed in the centre; if not, cook for a little longer before testing again. When ready, cool for five minutes before removing the cake from the tin and cooling completely on a wire rack.
For the icing:
For the icing, place all the icing ingredients in a bowl and beat together until light and fluffy. Cover the bowl with cling film and transfer to the fridge for 1-2 hours, until you are ready to ice the cake.
For the sugared carrots:
For the sugared carrots, place the carrots in a saucepan and just cover with cold water. Add the salt, sugar and butter. Place over a high heat and bring to the boil. Boil for 5-6 minutes - the carrots will cook as the water evaporates and a glaze will form in the base of the pan. Toss the cooked carrots in the glaze and then remove them from the pan to a plate with a bit more caster sugar. Roll them in a little more sugar, and leave to cool.
When the cake is completely cold, spread the icing over the top of the cake. Just before serving, scatter a few pecan nuts over the top of the cake and garnish with the sugared carrots.