In a stand mixer, place the softened unsalted butter, granulated sugar, and lemon zest. Beat on medium speed until the mixture is light and fluffy, about 3-5 minutes.

Once the butter mixture is fluffy, add in the sourdough starter, salt, and vanilla extract. Mix on low speed until everything is well combined, about 1-2 minutes.

Gradually add the all-purpose flour to the wet mixture, mixing on low speed until just combined.

Turn the dough out onto a clean surface and shape it into a cylinder about 14 inches long. Roll the dough tightly and coat it in granulated sugar.

Wrap the dough cylinder in plastic wrap and chill in the refrigerator for at least 30 minutes.

While the dough chills, preheat your oven to 375°F (190°C).

After chilling, unwrap the dough and slice it into 1/3 inch thick pieces. Bake for 12-15 minutes until the edges are slightly golden.

Allow the cookies to cool on the baking sheet for about 5 minutes before transferring to a wire rack.

While the cookies cool, whisk together melted butter, powdered sugar, lemon juice, and lemon zest until smooth.

Dip half of each cooled cookie into the glaze and let them set at room temperature.