In a bowl, beat the room temperature butter until smooth. Add the sugar and mix until combined. Add the egg, vanilla extract, and salt, and mix until well incorporated.
Gradually add the flour while mixing on low speed with a dough hook attachment until the dough comes together and is not sticky. If you are not using a mixer, start mixing with a wooden spoon and finish mixing with your hand.
Place the dough between two sheets of baking paper and roll it out to a thickness of 4-5mm. Remove the top baking paper and trim the dough into a rectangle. Re-roll any excess dough as needed.
Brush the rolled dough with melted butter.
Combine the granulated sugar and cinnamon in a small bowl. Sprinkle this mixture evenly over the buttered dough.
Using the bottom baking paper for support, carefully roll the dough into a tight log starting from one long edge. Wrap the log in plastic wrap to secure and chill in the fridge for 30-40 minutes.
Preheat the oven to 165°C (329°F) and line a baking tray with parchment paper.
Slice the chilled dough log into 2cm thick rounds and place them on the prepared baking tray.
Bake for 10-11 minutes, or until the edges are lightly golden. Remove the cookies from the oven and let them cool for a few minutes on the tray before transferring them to a wire rack to cool completely.
For the icing, whisk powdered sugar and water together until smooth. Drizzle over the cooled cookies using a spoon or piping bag.
Let the icing set before serving.