In your food processor, pulse the flour, powdered sugar, freeze dried strawberries and salt, until combined.With the processor running, drop in the pieces of butter, one at at time, pulsing a few times until the butter is evenly distributed, and the dough is moistened and crumbly. Add the vanilla and the food coloring, and pulse a few times.Add the milk, then process until the dough starts to stick together.
Dump the dough out onto a clean counter and use your hands to finish bringing the dough together, kneading in any stray bits of flour. The dough should be soft, smooth and supple.
Shape the dough into a flat disk, wrap in plastic wrap, and refrigerate for 20 minutes.Prepare two baking sheets by lining them with sheets of parchment paper.
Roll and Cut the Cookies
On a lightly floured surface, place the disk of dough, and dust the top of the dough with flour, too. Roll out to an even thickness of between 1/8 - 1/4 inch thick.Use mini flower shaped cookie cutters to cut as many cookies as you can, placing the cut cookies 1 inch apart on the parchment lined baking sheets. Using 1 1/2 inch mini cutters, you'll get about 74 small cookies (32 filled mini sandwich cookies)Gather up the scraps of dough, roll them out again, and continue cutting as many cookies as you can.
Place the baking sheets with the cut cookies in the refrigerator and chill for 30 minutes. Chilling the cut cookies helps to ensure they don't spread and hold their shape perfectly while baking.
Bake the Cookies
Preheat the oven to 350F, and position a rack in the center of the ovenBake the chilled cookies on the center oven rack for about 8 minutes. The tops should be a very pale golden with no dark browning, and you should see tiny flaky layers around the edges. If you (very carefully) turn a cookie over with a spatula, the bottom should have light golden browning, with no appearance of raw or wet dough.The bake time will completely depend on the width and thickness of your cut cookies. Larger/thicker cookies will take longer to bake. Be sure your cookies are fully baked.
Cool the cookies for 5 minutes on the pan, then very gently transfer them to a cooling rack to cool completely.Shortbread cookies are delicate when warm, so handle with care. The cookies will initially be slightly crisp around the edges, but they soften as they're stored and will remain wonderfully soft and buttery.
Let your cookies cool completely before frosting and decorating. You can bake the cookies in advance, store in an airtight container, and freeze for 3-6 months until you're ready to fill them.
Make the Buttercream
With an electric mixer (stand or hand held), beat the butter for 1 minute on medium speed until smooth.
Add the powdered sugar, meringue powder and lemon zest/extract. Beat for several minutes on medium speed until light and fluffy.
Fill the Cookies
Fit a 12 inch disposable piping bag with tip #32, or another small/medium star tip.
Turn half the cookies over, so the bottoms are facing up. Pipe the buttercream onto the cookies, then top with a second cookie and press down gently so you have a sandwich cookie.
Store the filled cookies in an airtight container in the refrigerator for up to 2 weeks, or in the freezer for up to 3-6 months.