Preheat oven to 350 degrees Fahrenheit and prepare baking sheets with nonstick liners or lightly grease them.
In the bowl of an electric mixer, cream together sugar, butter, egg and peppermint extract. In a separate bowl, stir together flour, baking soda, cream of tartar and salt. Add to butter mixture and mix until incorporated. Stir in 1 c. chopped white chocolate peppermint chunks by hand.
Scoop dough into 1 1/2" balls and place on cookie sheet. Be sure to only do 8 cookie balls per sheet-2 in each row-so they can have enough room to spread out.
Use a fork to press each dough ball down in a criss-cross manner. Bake for exactly 8 minutes, then remove from oven and allow to cool on cookie sheet for a few minutes before carefully removing to a cooling rack. After cookies are cooled, melt 3/4 cup white chocolate peppermint chunks (more if desired) and drizzle on top. You could also dip half of each cookie in the melted white chocolate peppermint for a different look. This would of course require more white chocolate peppermint chunks. Let chocolate set, then store in an airtight container.