In your food processor, pulse the flour, powdered sugar 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 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.
Cut and Chill the Cookies
Prepare two baking sheets by lining them with sheets of parchment paper.
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 between 1/8 - 1/4 inch thick.
Use a 4-inch bunny cookie cutter to cut as many cookies as you can, placing the cut cookies 2 inches apart on the parchment lined baking sheets. Gather up the scraps of dough, roll them out again, and continue cutting as many cookies as you can. With a 4 inch cookie cutter, you should get around 2 dozen cookies.
Place the baking sheets with the cut cookies in the refrigerator and chill for 1 hour - 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 350 F, and position a rack in the center of the oven.
Bake one pan of cookies at a time, keeping the other pan refrigerated until needed. Bake the chilled cookies on the center oven rack for about 11 1/2 - 12 minutes. The tops should be a very pale golden with no dark browning, and you should see tiny flaky layers around the edges.
Cool the cookies for 5 minutes on the pan, then very gently transfer them to a cooling rack to cool completely before frosting and decorating. Sugar cookies are delicate when warm, so handle with care.
Buttercream
Make 1 recipe Vanilla Buttercream, if you only want a thin layer of frosting on your cookies, or you can make 2x the recipe if you like more frosting.
Use a small offset icing spatula to gently frost the top of the cookies.
Before the frosting crusts over, sprinkle the cookies with coconut flakes and lightly press them into the frosting.
Notes
Making the Dough without a Food Processor: If you don't have a food processor, you'll need a large bowl and a pastry cutter. In the bowl, combine the flour, sugar and salt. Scatter with the cold butter pieces and use the pastry cutter to cut the butter in until very finely cut and evenly distributed. Drizzle with the vanilla and milk, then use your hands to work the liquid into the dry ingredients until it forms into a soft dough.
Storing the Baked Cookies: The baked cookies should be cooled completely then stored in an airtight container for up to 5 days. If you won't be frosting and decorating them immediately, freeze the cookies in an airtight container for 3-6 months.
Freezing the Dough: To make the dough in advance, wrap the disks of dough in plastic wrap, label them, and refrigerate for up to 5 days, or freeze for 3-6 months. Frozen dough should be thawed overnight in the refrigerator. Then take the dough out of the fridge about 1 hour before you're ready to roll out the dough. It should still be cool to the touch when you roll it out, but pliable enough to roll without too much difficulty.
Coconut: You can use either sweetened and shredded coconut, or unsweetened "natural" coconut flakes. Natural coconut is my preference. I also recommend pulsing the coconut a few times in a small food processor or ninja to chop it more finely, before sprinkling it onto the cookies.
Keyword Bunny, Coconut, Easter, High Altitude, Sugar Cookies