In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugar on medium-high speed for 1-2 minutes, until light and fluffy.
Reduce the mixer speed to low and add the eggs, one at a time, mixing until just incorporated after each addition.
Add the peppermint extract and green food coloring (if using). Mix until evenly combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, cream of tartar, and salt.
With the mixer on low speed, gradually add the dry ingredients to the wet ingredients. Mix until just combined.
Add the chocolate chips and chopped chocolate. Use a spatula to gently fold until evenly distributed.
Cover the dough and refrigerate for 1 hour.
Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.
Scoop about 2 tablespoons of dough and roll into balls. Place them about 2 inches apart on the prepared baking sheets.
Bake for 8-10 minutes, or until the edges are lightly golden and the centers are set.
Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.