To the bowl of a stand mixer fitted with the paddle attachment, add butter and granulated sugar. Beat on medium-high speed until light and fluffy, about 1 to 2 minutes.
With the mixer on low speed, add in the eggs, one at a time, mixing until just incorporated after each addition.
Turn off the mixer and add peppermint extract and green food coloring. Mix on low speed until evenly combined.
In a separate medium bowl, whisk together the flour, baking powder, baking soda, cream of tartar, and kosher salt.
With the mixer on low speed, gradually add the dry ingredients to the wet ingredients. Mix until just combined, then remove the bowl from the mixer.
Add the chocolate chips and finely chopped chocolate to the dough. Use a spatula to mix by hand until the chocolate is evenly distributed.
Cover the dough and refrigerate for about 1 hour.
Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper and set aside.
Using a medium cookie scoop (about 2 tablespoons), scoop the dough and roll it into balls. Place the dough balls about 2 inches apart on the prepared baking sheets.
Bake for 8 to 10 minutes, or until the edges are just starting to turn golden brown and the centers look set.
Allow the cookies to cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.