Prepare the vanilla dough by beating the butter and sugar in a medium bowl until soft and creamy. Add the vanilla, egg, and flour into the creamed mixture and stir until it forms a soft dough. Repeat and prepare the chocolate dough in the same way. Knead each dough separately until smooth, cover in plastic wrap and chill for 30 minutes.
Roll out both pieces of dough to 25 x 18-cm (10 x 7-inch) rectangles. Brush a little milk or egg white over the vanilla dough and lay the chocolate dough on top. Press down gently. Roll up the dough from the short side to make a spiral log. Wrap and chill for 30 minutes or freeze until required. If frozen, thaw overnight before continuing with the next step.
Preheat the oven to 180 degrees Celsius. Line baking sheets with parchment paper. Cut the roll into 24 to 30 slices, arrange 4-cm (1 1/2-inch) apart. Bake for 10 to 12 minutes until pale golden. Remove from the oven and cool for 5 minutes. Transfer to a wire rack to cool completely.