1 3-inch round cookie cutter, 5 to 6 pastry bags fitted with tiny round or star tips, decorative cookie cutters (optional)
For the cookies: In a medium bowl, combine the flour, lavender, baking powder and salt and set aside.
In a stand mixer fitted with a paddle attachment or in a large bowl with an electric mixer beat together the butter and both sugars until pale and fluffy, about 5 minutes. Beat in the lemon zest. Add the eggs, one at a time, beating after each, and then beat in the vanilla. Add the flour mixture to the butter mixture, mixing on low until blended.
Turn the dough out onto the counter and divide into 2 large discs. Wrap each in plastic wrap and refrigerate for 1 hour up to 2 days.
When ready to bake preheat the oven to 375 degrees F and line 2 baking sheets with parchment or a silicone mat.
Working with 1 dough disc at a time, roll it out on a lightly floured surface to 1/2-inch-thick. Use a 3-inch round cookie cutter to cut out circles. Transfer to the prepared baking sheets spacing 1-inch apart. Gather and re-roll the scraps and cut out more circles.
Bake until the cookies are just thinking about starting to turn brown, start checking at around 12 minutes. Let cool on the baking sheets for 5 minutes and then transfer to a wire rack to cool completely.
Meanwhile, for the buttercream: In a stand mixer fitted with a paddle attachment or in a large bowl with an electric mixer, beat together the powdered sugar and butter until smooth. Divide the buttercream amongst individual bowls (I suggest 5 or 6) and stir food coloring into each. Transfer to pastry bags fitted with tiny round or star tips.
For the glaze: Whisk together the powdered sugar, corn syrup, milk, vanilla, salt and a few drops of food coloring, if using, until very smooth. You're going for the consistency of a thick glue. If it's too thick-- add a couple more drops of milk and if it's too thin-- add a few spoonfuls of powdered sugar.
Assemble the cookies: Dip the tops of each cooled cookie into the glaze, letting any excess drip off, and then place on a wire rack to dry. After a few minutes, once they're mostly dry to the touch (completely dry isn't necessary) you can go ahead and pipe on buttercream decorations. Go the freehand route or use cookie cutters to press indentations into the glaze which can serve as an outline for your decorations.