Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.

To make the cookies: In a medium bowl, beat together the butter and sugars.

Add the egg, beating well and scraping the bottom and sides of the bowl.

Add the peppermint, beating to combine.

Add the flour, cocoa, baking soda, and salt, beating to combine. Taste the dough; add more peppermint, if desired.

Stir in the chocolate chips.

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.

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.

Bake the remainder of the cookies, letting them cool right on the pans, or transferring to a rack if you need the pans.

When the cookies are completely cool, make the filling: stir together the sugar and soft butter until thoroughly combined. Add the half & half, stirring to make a spreadable filling. Adjust the consistency with extra half & half or confectioners' sugar, if necessary.

To make the coating: Melt the chocolate slowly and gently, either in a microwave, or in a double boiler set over a burner.

Dip half of each cookie into the chocolate. Alternatively, drizzle the chocolate over the tops of the cookies.