Preheat the oven to 350°F and line a baking sheet with parchment paper.
In a mixing bowl, cream the softened butter, brown sugar, and granulated sugar together until the mixture becomes lighter and fluffy, which usually takes about two minutes.
Add the sourdough discard, egg, vanilla extract, and white vinegar, then mix until everything is fully combined.
In a separate bowl whisk together the flour, cocoa powder, baking soda, cream of tartar, and salt.
Add the dry ingredients to the wet ingredients and mix just until a soft dough forms. Do not overmix. Fold in the white chocolate chips if you are using them.
Scoop the dough into balls about 2 tablespoons each and place them on the baking sheet with space between them.
Bake for 13 to 15 minutes, until the edges are set but the centers still look slightly soft. Let the cookies rest on the baking sheet for about five minutes before transferring them to a rack to cool.
This dough can also be scooped and refrigerated for up to 24 hours before baking, which usually makes the cookies thicker and gives them even better flavor.