Sift together flour, baking powder, and salt. Set aside. Place butter and sugar in large bowl of electric stand mixer and beat until light in color. Add egg and milk and beat to combine. Put mixer on low speed, gradually add flour, and beat until mixture pulls away from the side of the bowl. Divide the dough in half, wrap in waxed paper, and refrigerate for 2 hours.
Preheat oven to 375º F.
Sprinkle surface where you will roll out dough with powdered sugar. Remove 1 wrapped pack of dough from refrigerator at a time, sprinkle rolling pin with powdered sugar, and roll out dough to 1/4-inch thick. Move the dough around and check underneath frequently to make sure it is not sticking. If dough has warmed during rolling, place cold cookie sheet on top for 10 minutes to chill. Cut into desired shape, place at least 1-inch apart on greased baking sheet, parchment, or silicone baking mat, and bake for 7 to 9 minutes or until cookies are just beginning to turn brown around the edges, rotating cookie sheet halfway through baking time. Let sit on baking sheet for 2 minutes after removal from oven and then move to complete cooling on wire rack.
Serve as is or ice as desired. Store in airtight container for up to 1 week.
Add powdered sugar, meringue powder, lavender vanilla simple syrup, and 7 Tbsp. of water to a large bowl. Using a hand mixer or stand mixer fitted with a whisk attachment, beat ingredients together on high speed for 1.5- 2 minutes. When lifting whisk off icing, it should drizzle down and smooth out within 5-10 seconds. If it’s too thick, beat in more water 1 Tbsp. at a time.
When applied to cookies or confections in a thin layer, icing completely dries in about 2 hours at room temperature. If icing consistency is too thin and runny, it will take longer to dry. If the icing is applied very thick on cookies, it will also take longer to dry. Place cookies in the refrigerator to help speed up drying process.
Garnish with lavender buds while icing is still setting up.