1. In a small saucepan or cast iron pan, melt butter over medium heat.
2. Stir the butter as it melts for about 10 minutes. The butter will begin to foam and turn a deep golden color. When the butter has finished browning, the foam will have subsided and there will be milk solids at the bottom of the saucepan.
3. Allow the browned butter to come to room temperature.
4. Preheat the oven to 375 degrees F. Position an oven rack in the middle of the oven.
5. In a medium-sized bowl, combine the browned butter with the sugar, vanilla, and egg yolk. Mix together with a fork.
6. Sift the cake flour and cream of tartar together, directly into the bowl with the other ingredients.
7. Using a large wooden spoon, mix the ingredients together until a stiff dough forms.
8. Cover a baking sheet with parchment paper or grease and flour it.
9. Using your hands, roll the dough into walnut-sized balls. Space the balls of dough over the baking sheet, leaving a walnut-sized space between each ball.
10. Bake the cookies for 15 to 20 minutes or until golden.
11. Let the cookies cool for 5 minutes before transferring to a cooling rack (or eating immediately!)