Begin by creaming the unsalted butter and powdered sugar together in a large mixing bowl using an electric mixer. Beat the mixture on medium speed until it becomes light and fluffy, which should take about 1 minute.
Once the butter and sugar are creamed, add the freshly grated Meyer lemon zest and vanilla extract to the mixture. Mix at low speed until everything is well combined.
In a separate bowl, whisk together the all-purpose flour, tapioca starch, and salt. Make sure to mix well to evenly distribute the salt and starch throughout the flour.
Gradually add the dry mixture to the creamed butter mixture, beating on low speed until a soft dough forms. Aim for a cohesive texture without any dry flour visible.
Transfer the dough onto a wax paper-lined surface, pressing it down evenly to form a rectangle about 12x4 inches and ¼ inch thick. Wrap the dough in wax paper and refrigerate for at least 1 hour.
Preheat your oven to 350°F (175°C). Remove the dough from the refrigerator and cut it into 1x1.5-inch rectangles. Place these on parchment-lined baking sheets.
Bake the cookies for 12-15 minutes, or until the edges are barely golden. Let the cookies cool on the baking sheet for 15-20 minutes.
While the cookies cool, prepare the icing by whisking the sifted powdered sugar, fresh Meyer lemon juice, and milk together until you achieve a smooth consistency.
Once the cookies have cooled, drizzle the prepared icing over each one and garnish with long strips of Meyer lemon zest.