Beat the butter and icing sugar together until smooth.
Push the hard-boiled egg yolk through a sieve and stir the raw egg yolk, lemon and vanilla into it. Add this to the butter mixture and stir until blended.
Add the flour and salt to the butter mixture and stir until blended. Shape the dough into logs that are an inch across and wrap in plastic wrap. Chill until firm, about 2 hours.
Preheat the oven to 180°C and line 2 baking trays with parchment paper.
Pour the demerara sugar onto a plate. Unwrap the cookie logs and roll them in the sugar to coat. Slice the cookies about 1/4-inch thick and place on the baking trays, leaving a 1/2-inch between them. Bake the cookies for about 15 minutes, until lightly browned on the bottoms. Cool the cookies on the tray and then store in an airtight container.
The cookies will keep for up to 4 days in an airtight container.