I had a quick look in the book but no recipe!! Is the author referring to a brand or just one that does not use honey, sugar, nuts etc?? I am confused too, but wanted to give you answer! Here is a yummy carrot cake recipe if you need one:

Ingredients (serves 12)
Olive oil, to grease
2 (about 300g) carrots
1 cup (150g) self-raising flour (wholemeal would be okay)
1/2 cup (75g) plain flour
1 tsp bicarbonate of soda
1/2 tsp ground cinnamon
1/2 cup (80g) brown sugar (reduce if you like??)
3/4 cup (185ml) oil (reduce if you like??)
1/2 cup (125ml) golden syrup (reduce if you like??)
3 eggs
1 tsp vanilla essence

Icing
250g spreadable cream cheese
1/2 cup (80g) icing sugar
1/2 tsp vanilla essence
Method
Preheat oven to 170?C. Grease a 20cm (base) round cake pan lightly with oil, and line with non-stick baking paper. Peel and grate the carrots, and set aside. Sift the flours, bicarbonate of soda and cinnamon into a large bowl.
Put the brown sugar, oil, golden syrup, eggs and vanilla in a separate bowl. Use a balloon whisk to mix until combined.
Pour the oil mixture into the dry ingredients. Use a wooden spoon to stir gently until just combined. Stir in the grated carrot.
Pour the mixture into the pan and bake for 1 hour. Set aside for 5 minutes, before turning out onto a wire rack to cool completely.
To make the icing, place the cream cheese, icing sugar and vanilla in a bowl. Use a wooden spoon to mix until well combined.
Spread the icing over the cake.

There is hardly any sugar in it (relatively speaking about cakes here!) and I would use a bit less oil- maybe half a cup.
I made a chocolate mud cake for my DD's 1st birthday and she had never had sugar before and I only gave her a couple of little cubes. She actually didn't want anymore- not that she was getting much more anyway!
PS scrape the icing off her piece too to lessen the sweetness for her.