Preheat an oven to 350 F. Line a baking sheet with parchment paper and set aside.
Place the candy canes in a ziplock bag and seal. Use a rolling pin to smash them into small pieces.
Cream the butter, cream cheese, and sugars. In a mixing bowl, use a hand mixer to cream the butter, cream cheese, and both sugars together until light and fluffy. Mix the eggs in one and a time and mix in the vanilla.
Fold in the dry ingredients. In a separate bowl, whisk together the flour, baking soda, cream of tartar, and salt together. Add the dry ingredients to the wet and mix until just barely combined.
Mix in the chocolate and peppermint. Using a rubber spatula, fold in the chopped chocolate and candy canes until evenly dispersed.
Scoop. Use a cookie scoop to create uniform cookie dough balls so that they bake evenly. Use a 3 tablespoon scoop to scoop the cookie dough into balls. Arrange them on a parchment lined baking sheet, giving some space between each cookie.
Bake. Bake for 12-14 minutes, until puffy but still soft in the center. Let them cool and then drizzle with melted white chocolate, if desired.