In a large bowl or the bowl of a stand mixer fitted with the flat beater, beat the butter, confectioners’ sugar, vanilla, salt, and peppermint oil until just combined (not aerated), about 30 seconds on medium speed. Scrape the sides of the bowl. , Weigh your flour and your cocoa; or measure them by gently spooning it into a cup, then sweeping off any excess. Add the flour and cocoa and mix until a homogenous dough forms, about 30 seconds on low speed. Scrape the sides of the bowl. , Add the chocolate, white chocolate, and peppermint candy and mix until just combined, about 10 seconds on low speed. Scrape the sides of the bowl and stir the dough with a flexible spatula to ensure the mix-ins are well distributed. , Transfer the dough to a sheet of parchment and shape it into a log, roughly 1 ½" wide and 11" long. To make a more perfectly round circle, fold the parchment away from you to encase the log. Use your non-dominant hand to press a long, straight object (a ruler or a bench knife works well) in between the dough and the work surface, then use your dominant hand to gently tug the top piece of parchment away from you. This should pull the dough so that it butts up against the straight object and makes a very smooth, symmetrical log. Alternatively, press the dough into one half of the Slice-and-Bake Cookie Dough Keeper, using your hands to round the dough slightly into an even, smooth log. , Wrap the dough in plastic or your favorite reusable wrap, or cover it with the other half of the Cookie Dough Keeper, and refrigerate for at least 2 hours or up to 4 days. , When ready to bake, allow the dough to sit at room temperature for 30 to 45 minutes so that it warms up slightly and can be easily sliced. , Preheat the oven to 350°F with racks in upper and lower third. Line 2 baking sheets with parchment or silicone mats, or lightly grease. , Unwrap the dough (or remove it from the Cookie Dough Keeper) and use a serrated knife to cut the dough into ½"-thick slices. (Measure the first cookie with a ruler, as ½" is likely much thicker than you think, and slicing the dough too thinly may cause the cookies to crumble.) Distribute the slices evenly between the prepared baking sheets. , Bake the peppermint cookies for 12 to 14 minutes, rotating the pans halfway through baking. When fully baked, the cookies will be puffed and no longer look wet in the center. Allow the cookies to cool completely on the baking sheets before touching them; they're quite fragile when warm. (They will deflate somewhat while cooling.) , Store leftover peppermint cookies in an airtight container at room temperature for up to 1 week; freezer for longer storage. Store dough, well-wrapped, for up to 1 month in the freezer. Thaw in the refrigerator for 24 hours before planning to bake.