Preheat the oven to 350°F/175°C and line two large baking sheets with silicone mats or parchment paper. Set aside.
In a large bowl or the bowl of a stand mixer, beat 1/2 cup butter, 1/2 cup granulated sugar, and 1/2 cup packed light brown sugar on medium-high speed with a hand mixer or paddle attachment for a couple of minutes. The mixture should become lighter in color and look fluffy as you mix.
Add 1 egg, 1 large egg yolk, 1 Tbsp instant espresso or coffee, 1 tsp vanilla extract, and 1 tsp peppermint extract. Mix on medium speed until incorporated.
Whisk together 1 cup flour, 1/2 cup unsweetened cocoa powder, 1 Tbsp cornstarch, 1 tsp baking soda, and 1/2 tsp salt in a separate bowl.
Add the flour mixture into the butter mixture in two additions, mixing on a low speed until it is JUST incorporated. Scrape the sides of the bowl with a rubber spatula between additions.
Fold in 1 1/4 cups of milk or dark chocolate chips with a rubber spatula.
Use a cookie scoop to scoop 1 1/2 Tbsp-sized cookie dough balls. You should be able to scoop out about 24 cookies. Place the cookies on the prepared baking sheets and space them about 2 inches apart.
Bake one sheet of cookies at a time. Bake for 11–12 minutes on the middle rack of your oven (bake time can vary based on the size of the cookies). They will look kind of puffy coming out of the oven and deflate a bit as they cool, but this is what makes them so deliciously chewy! Let the baked cookies cool on the pan for about 15 minutes, then transfer to a wire rack to finish cooling.
While the cookies cool, melt the white chocolate chips in a heatproof bowl using a double boiler or microwave (melt in 15-second intervals, and stir in between until completely melted). Drizzle the melted white chocolate over each cookie and sprinkle the crushed peppermint candies on top of the chocolate. These will set at room temperature in about an hour. If you're in a rush, you can set them in the fridge to help the chocolate set faster. Leftover cookies can be stored at room temperature in an airtight container for up to a week.