In a large mixing bowl, beat butter until creamy. Add granulated sugar and brown sugar and beat until color is a pale yellow. Add egg, vanilla extract, and peppermint extract and incorporate into mixture. Set aside.
In a medium mixing bowl, whisk together flour, cocoa powder, baking soda, espresso powder, and salt. Slowly, add these ingredients into the wet ingredients. Add chocolate chips and mix thoroughly. Transfer dough into plastic wrap and refrigerate for at least three hours.
Preheat oven to 350 degrees Fahrenheit and place rack in the center of oven. Line 2 large cookie sheets with parchment paper and set aside. Remove cookie dough from refrigerator and allow to come to room temperature (about 20 minutes).
Scoop cookie dough (approximately 25 grams of dough per cookie) and roll into balls. Place onto baking sheet 2 inches apart.
Bake the cookies for approximately 9 minutes. Remove from the oven and cool on cookies sheet for 5 minutes. Remove to rack to cool completely.
Melt white chocolate in microwave in 10 second intervals. Dip half of chocolate cookie in melted white chocolate and place on silicone mat or waxed paper. Sprinkle with peppermint chips while chocolate is still "wet". Allow cookies to set at room temperature or in the fridge.