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Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.
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To make the cookies: In a medium bowl, beat together the butter and sugars.
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Add the egg, beating well and scraping the bottom and sides of the bowl.
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Add the peppermint, beating to combine.
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Add the flour, cocoa, baking soda, and salt, beating to combine.
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Stir in the chocolate chips.
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Drop a few chestnut-sized (about 1" or so) balls of cookie dough onto the prepared baking sheets, spacing them about 1 1/2" to 2" apart. A slightly heaped teaspoon cookie scoop works well here. Since these cookies are dark-colored, it's difficult to tell when they're done. We suggest you bake 2 or 3 cookies alone first, before baking a couple of baking sheets' worth. That way, you can make sure you bake them to just the right degree of doneness.
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Bake the cookies for 10 to 12 minutes (if they're on parchment-lined baking sheets); bake a couple of minutes less if they're on dark sheets without parchment. Remove them from the oven, and transfer them to a rack to cool. If they're crunchy and you prefer soft, bake the rest of the cookies a shorter amount of time; if soft and you're looking for a crunchy cookie, you'll know you need to bake them longer.
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Bake the remainder of the cookies, letting them cool right on the pans, or transferring to a rack if you need the pans.
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When the cookies are completely cool, make the filling: stir together the sugar and soft butter until thoroughly combined. Add the half-and-half, stirring to make a spreadable filling. Adjust the consistency with extra half-and-half or confectioners' sugar, if necessary.
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To make the coating: Melt the chocolate slowly and gently, either in a microwave, or in a double boiler set over a burner.
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Dip half of each cookie into the chocolate. Alternatively, drizzle the chocolate over the tops of the cookies.
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Store cookies, loosely wrapped, at room temperature for several days. Freeze for longer storage.
