Bring the water and sugar to the boil, then reduce the heat and simmer for approx. 5 mins. Pour the liquid into a measuring jug and allow to cool completely. Add the basil and lemon juice, puree until smooth. Strain the mixture into a stainless steel bowl.
Cover and freeze for approx. 3 hrs. Stir well with a whisk every hour.
Place the oil in a bowl, whisk briefly with the sugar, salt, lemon zest and egg white until smooth. Add the flour, mix quickly to form a soft dough – do not knead. Flatten the dough, cover and chill for approx. 30 mins. Roll out the dough between a folded sheet of baking paper to a thickness of approx. 1 cm. Cut out 4 circles (each approx. 4 cm in diameter), transfer to a baking tray lined with baking paper.
Bake for approx. 10 mins. in the centre of an oven preheated to 200°C. Remove from the oven, cool the biscuits on a cooling rack. Dust the lemon biscuits with icing sugar.