Preheat the oven to 175 C. Line a 9x5 inch loaf pan with non-stick baking paper, leaving a little overhang for easy removal.
In a mixing bowl, combine the sugar and orange zest and rub them together to release more flavour from the orange. Add the eggs and whisk until lighter and frothy, about a minute. A balloon whisk is great for this recipe.
Next, whisk in the vanilla extract, olive oil and yoghurt until smooth.
Sift in the flour, baking powder, baking soda and salt. Fold with a spatula to combine and if there are lumps, briefly run the whisk through the batter to smoothen it. Do not over-mix.
Pour into the prepared cake pan, smoothen the top and bake for 30 to 35 minutes until the top is golden-brown and a toothpick poked in the center comes out clean or with a few moist crumbs.
Allow the cake to cool in the pan for 15 minutes then lift out using the paper overhang and let it cool completely on a wire rack.
To make the glaze, sift the icing sugar into a small bowl, add the orange juice and whisk till smooth and slightly runny. See notes for a thicker glaze.
Pour slowly over the cooled cake and let it rest for about 20 minutes. The glaze will soak slightly into the top of the cake.
Slice into 8 to 10 thick slices and cut each one further for smaller servings if desired. This cake is a little fragile since it's so moist, so make sure to cut it only when it's cool.
Store at room temperature for 3 to 4 days in an airtight tin. If it's warm where you live, this is best stored in the fridge after the first couple of days where it will keep for a week at least. I actually love oil-based cakes when cold from the fridge! Happy baking :)