Cream the Butter and Sugars: In a large mixing bowl, combine ½ cup softened unsalted butter, ¾ cup packed light brown sugar, and ½ cup granulated sugar. Beat with an electric mixer on medium speed for approximately 2–3 minutes, until the mixture looks creamy, slightly lighter in color, and evenly blended. Scrape down the sides and bottom of the bowl as needed. The mixture should be smooth and fluffy rather than greasy or melted.

Add the Egg and Flavorings: Add 1 large egg, 1 teaspoon pure vanilla extract, and ¼ teaspoon peppermint extract. Beat on low to medium speed until fully incorporated. The mixture should look smooth and cohesive, with no visible streaks of egg. Avoid adding extra peppermint extract at this stage because its flavor becomes more noticeable after baking.

Combine the Dry Ingredients: In a separate medium bowl, whisk together 1½ cups all-purpose flour, ½ cup unsweetened natural cocoa powder, ½ teaspoon baking soda, and ½ teaspoon fine sea salt. Whisk until the mixture is evenly colored and no large cocoa lumps remain. This helps distribute the leavening evenly and prevents dry pockets in the finished cookies.

Prepare the Chocolate Cookie Dough: Add approximately half of the dry mixture to the butter mixture and mix on low speed until partially combined. Add 2 tablespoons whole milk, followed by the remaining dry ingredients. Mix just until a thick, soft chocolate dough forms. The dough should be cohesive, slightly tacky, and firm enough to hold its shape when scooped. If the dough looks dry at first, continue mixing briefly before making any adjustments. The flour and cocoa may need a few moments to absorb the moisture.

Chill the Dough: Cover the bowl and refrigerate the dough for 30 minutes. Chilling helps firm the butter and allows the flour and cocoa powder to hydrate. The dough should feel slightly firmer and less sticky after chilling. If it becomes too hard to scoop, leave it at room temperature for a few minutes before shaping.

Preheat the Oven and Prepare the Baking Sheets: Near the end of the chilling time, preheat the oven to 350°F (175°C). Position a rack in the center of the oven. Line two large baking sheets with parchment paper. Avoid greasing the baking sheets heavily because excess fat can encourage the cookies to spread.

Scoop and Shape the Cookies: Use a 1½-tablespoon cookie scoop to divide the dough into 24 evenly sized portions. Roll each portion gently between your palms to create a smooth ball. Arrange 12 dough balls on each baking sheet, leaving approximately 2 inches (5 cm) between them. Press each ball down slightly to create a thick disk. The dough should remain relatively thick so the centers stay soft during baking.

Bake the Chocolate Cookies: Bake one sheet at a time for approximately 10–12 minutes. The cookies should look set around the edges, with centers that are slightly soft but no longer wet or glossy. Because the dough is dark, color alone is not a reliable indication of doneness. Look for a slightly puffed surface with small cracks and edges that hold their shape. Avoid overbaking, as chocolate cookies can become dry once they cool.

Cool the Cookies Completely: Remove the baking sheet from the oven and allow the cookies to rest on the sheet for approximately 5 minutes. They may feel delicate immediately after baking but will firm up as they cool. Transfer them carefully to a wire rack and let them cool for approximately 15 additional minutes. Make sure the cookies are completely cool before dipping them into white chocolate. Warm cookies can soften the coating and make it difficult to achieve a neat finish.

Crush the Peppermint Candy: Place the candy canes or peppermint candies in a resealable food-safe bag. Seal the bag and gently crush the candy with a rolling pin or another heavy utensil. Aim for a mixture of small pieces and fine crumbs. Avoid leaving very large chunks, which can make the cookies difficult to bite. Measure ⅓ cup of crushed candy and set it aside for decorating.

Melt the White Chocolate: Place 10 ounces white chocolate melting wafers or finely chopped white chocolate in a clean, completely dry microwave-safe bowl. Microwave at 50% power in 15–20-second intervals, stirring thoroughly between each interval. Stop heating when most of the chocolate has melted, then continue stirring until smooth. White chocolate can scorch more easily than dark chocolate, so avoid overheating it. If the coating is too thick for dipping, add the optional 1 teaspoon refined coconut oil and stir until fully incorporated. The melted chocolate should be glossy and fluid enough to coat the cookies without forming a heavy layer.

Dip the Chocolate Cookies: Hold a completely cooled cookie by one edge and dip approximately half of it into the melted white chocolate. Lift it slowly and allow excess chocolate to drip back into the bowl. Gently scrape the bottom edge against the bowl if necessary. Place the dipped cookie onto a parchment-lined baking sheet. Repeat with the remaining cookies. If the chocolate begins to thicken, warm it briefly at low power and stir until smooth again.

Add the Crushed Peppermint Topping: While the white chocolate coating is still wet, sprinkle the dipped portion of each cookie with crushed peppermint candy. If desired, add a few of the optional white chocolate curls for an extra decorative finish. Press the peppermint pieces very gently if needed so they adhere to the coating. Avoid adding too much candy, which can overwhelm the chocolate flavor and make the topping difficult to keep in place.

Let the White Chocolate Set: Leave the dipped cookies undisturbed at a cool room temperature for approximately 30 minutes, or until the coating feels firm. The setting time depends on the type of white chocolate and the temperature of the room. Melting wafers generally firm up more reliably than untempered real white chocolate. If necessary, refrigerate the cookies briefly to help the coating set, but avoid prolonged chilling that may cause condensation to form on the candy topping.