In a food processor, pulse the flour, cocoa powder and powdered sugar to combine.With the food processor running, drop the cold butter in, one piece at a time, pulsing until well distributed.Add the vanilla and milk. Run the food processor until the mixture comes together into a soft dough. Turn the dough out, and knead 3-4 times to bring together any stray floury bits. Shape the dough into a flat disk, wrap in plastic wrap, and refrigerate for 20 minutes.
Alternatively, if you don't have a food processor, you can make the dough in a bowl, using a pastry cutter. Using a food processor is quick and easy, but I've made the dough by hand many times as well.
Line a large baking sheet (or two medium sized) with a sheet of parchment paper. Have a ruler and a pastry cutter or sharp knife ready. Get your letter embossing cookie stamps ready, too, if you're using them.
Set the chilled dough on a lightly floured surface, and lightly flour the top of the dough as well. Roll the dough out to an even thickness; it should be between 1/4 - 1/8 inch thick (1/4 inch is too thick, but you don't want to go as thin as 1/8 inch).
Using the ruler as a straight edge for gliding your pastry cutter against, cut the dough into strips that are 1 1/4 inch wide. Cut 4-inch lengths from each strip. Your cookies should measure 1 1/4 x 4 inches.
Set the cut cookies 2 inches apart on the parchment-lined baking sheet.Gather up the scraps of dough, knead together, re-roll, and finish cutting as many cookies as you can. Once all your cookies are cut, organize your letter embossers into short words and phrases. Dust them lightly with flour, and then press them into your cookies.Be sure to poke a hole at the top of your cookies, for your twine. I used the small end of a #4 piping tip. Refrigerate the cut cookies for 1 hour. Chilling them helps ensure they don't spread while baking.
Preheat the oven to 350. Bake the chilled cookies on the center oven rack for about 10-12 minutes. The only way to really tell if chocolate sugar cookies are done, is to look at the underneath. The cookies are extremely fragile while hot, so if you're not sure they're done, very carefully turn one over; if the dough looks damp underneath, bake the cookies for a couple more minutes.
Cool for 10 minutes on the baking sheet, then carefully transfer them to a cooling rack to cool completely.
Line a baking sheet with a piece of wax paper or parchment paper to set the dipped cookies on.Cut lengths of twine, about 6 inches long. Carefully thread the twine through the holes in the cookies. You can tie a knot, or a bow, whatever you like. Melt the white chocolate and coconut oil in your microwave in a 1-cup glass measuring cup, for 2-3 minutes at 50% power. Stir until smooth.
Gently dip the cookies into the white chocolate. Let the excess drip off, then set on the baking sheet.When dipping your cookies, I'd advise holding them in the middle, and not by the end. If you hold them by the end, the weight of the chocolate at the other end can break them in half. Before the chocolate hardens, sprinkle with the crushed peppermint.Let the chocolate set and harden completely before storing the cookies.