Place the butter in a large mixing bowl. Cream with an electric mixer until creamy. While the mixer is running, add in the sugar gradually and beat until the mixture is light, pale in colour, creamy and fluffy. Add in vanilla (if making vanilla dough) and the egg yolk and beat until the egg yolk has been incorporated. Add in the flour and cocoa/matcha powder (if making other flavoured doughs). Stir with a spatula until all the flour has been incorporated and a smooth and soft dough has formed.
Turn the dough out on a large piece of clingfilm. Flatten the dough into a rectangle and wrap it with the clingfilm. Refrigerate and chill for at least 30 minutes until the dough is firm.
Repeat to make other doughs.
Once the dough has chilled, cut the doughs into half. Roll each half of the dough out into a 5 x 10-inch rectangle. Brush with a thin layer of milk and lay your desired mix-and-match flavour on top. Roll it up tightly into a log. Wrap the log with clingfilm and refrigerate again for at least 30 minutes.
Preheat the oven to 180 degrees Celsius. Line a baking sheet with parchment paper and set aside. Slice the log into 1/4-inch slices and arrange them on the baking sheet, spacing them 1 1/2-inch apart (they will expand during baking). Bake for 10 to 12 minutes until lightly golden in colour. Cool the cookies on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
Store the cookies in an air-tight container for up to 1 week.