Begin by browning the butter in a saucepan over medium heat until it turns golden brown and emits a nutty aroma, then let it cool to room temperature.
In a mixing bowl, combine the cooled browned butter with powdered sugar and vanilla extract, mixing until smooth and creamy.
Gradually add the all-purpose flour and a pinch of salt, mixing until a dough forms.
Divide the dough into two portions and shape each into a log about 1.5 inches in diameter. Wrap the logs in plastic wrap and refrigerate for at least 2 hours, or until firm.
Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
Roll the chilled dough logs in granulated sugar for an extra sweet crunch on the outside.
Slice the logs into 1/4-inch thick rounds and place them on the prepared baking sheet, spacing them about an inch apart.
Bake for 12-15 minutes, or until the edges are lightly golden.
Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.