In a mixing bowl, combine the softened butter and powdered sugar. Beat them together until the mixture becomes light and fluffy.
Add the lemon zest, lemon juice, and vanilla extract to the mixture of butter. Mix well until fully combined.
In a separate bowl, whisk together the all-purpose flour and salt. Slowly incorporate the flour mixture into the butter mixture, mixing until a soft dough is formed.
Divide the dough into two equal portions. Shape each portion into a log about 1.5 inches in diameter. Wrap the dough logs tightly in plastic wrap and place them in the refrigerator for a minimum of 1 hour, or until they become firm.
Preheat your oven to 350°F (175°C) and prepare a baking sheet by lining it with parchment paper or a silicone baking mat.
Unwrap the chilled dough logs and slice them into 1/4-inch thick rounds. Place the rounds onto the prepared baking sheet, leaving a little space between each cookie.
Bake the cookies for 12-15 minutes, or until the edges turn slightly golden. Be careful not to overbake.
Remove the cookies from the oven and let them cool on the baking sheet for a few minutes, then transfer them to a cooling rack.
Once the cookies have cooled slightly, roll them in powdered sugar to coat them evenly. Allow them to cool completely before serving.