Preheat your oven to 375°F. Prepare two baking sheets by lining them with parchment paper or silicone baking mats.
In a mixing bowl, beat together the butter, salt, vanilla extract, and baking powder until the mixture is smooth. Add sugar and continue beating until creamy. Next, beat in the egg until fully incorporated.
(Ingredients: ⅓ cup salted butter, ½ teaspoon salt, 1 teaspoon vanilla extract, 1 teaspoon baking powder, ½ cup granulated sugar, 1 large egg)
Add flour and milk alternately to the mixture, doing so in two or three additions, mixing gently after each addition just until combined. Avoid overmixing the dough.
(Ingredients: 1 ½ cups all-purpose flour, ⅓ cup milk)
Use a small cookie scoop (approximately 1 tablespoon of dough per cookie) to portion out the dough onto the prepared baking sheets. Space cookies 2-3 inches apart to allow spreading during baking. Slightly wet your fingers or use the bottom of a glass to gently press each cookie into a circle about 1½ inches wide.
Bake the cookies for 9-11 minutes. They're ready when lightly golden on the bottoms, still pale on top, with just a slight browning around the edges. Take care not to overbake.
Prepare the Frosting: Combine all frosting ingredients in a double boiler, stirring occasionally until smooth and combined.
Flip the cookies upside down on the baking sheets. Spoon about 2 tablespoons of frosting onto the flat side of each cookie, forming a mound. Allow the cookies to sit at room temperature until the frosting fully sets, approximately 3 hours. Once set, store cookies in an airtight container at room temperature for up to 3 days.