In a large bowl, with an electric mixer, beat the butter with the sugar until fluffy. Add the egg, peppermint extract and vanilla, and continue beating until blended and smooth.
In a bowl, stir together the flour and cocoa powder. Add this dry mixture to the butter mixture, beating just until it forms a sticky dough. On two sheets of waxed paper or plastic wrap, have your young helper form the dough into two equal logs, about 1½ in. (4 cm) thick. (Roll it lightly on the table so that it is firmly packed and even on the surface.) Wrap tightly and refrigerate until chilled and firm - at least 2 hours or overnight.
When ready to bake, preheat the oven to 350°F (180°C). Line a baking sheet (or two) with parchment paper or grease lightly.
Unwrap a log of dough and cut it crosswise into ¼ in. (0.5 cm) thick slices. Place on the prepared baking sheet 1 in. (2 cm) apart (to allow for some spreading). Bake for 15 to 20 minutes, or until cookies are firm and dry to the touch (longer if you like them crisp). Let cool completely on racks.
Melt chocolate in a small saucepan set over another saucepan of boiling water (or in a double boiler, if you have one). Be careful not to scorch the chocolate or overheat. Line a baking sheet or rack with waxed paper.
Working with one cookie at a time, dip it about one-third of the way into the melted chocolate, then shake off the excess chocolate and place on waxed paper. Repeat until all the cookies are done. There should be enough chocolate to coat all the cookies, but if your child is helping you, there's no telling how quickly it will disappear. You may have to melt a couple of extra squares. If you don't feel like dipping cookies one by one, you can just lay them all out on a sheet of waxed paper and drizzle the melted chocolate over them all at once. Pretty cool too.
Let chocolate harden completely at room temperature (or in the refrigerator) before devouring cookies.
Sorry, there is no nutrition information available for this recipe.