In a small saucepan, brown the butter over medium-low heat until it turns a rich amber color and develops a nutty aroma. Remove from heat and immediately transfer to a small bowl. Let the brown butter cool until it solidifies. You can speed up the cooling process by refrigerating the bowl or placing it over another bowl filled with ice and stirring.
½ cup (113 grams or 1 stick) unsalted butter
In a small bowl, combine the flour, cornstarch, baking powder, baking soda, and salt. Set aside.
1 ¾ cups (220 grams) all-purpose flour, 1 tablespoon (8 grams) cornstarch, ½ teaspoon baking powder, ¼ teaspoon baking soda, ¼ teaspoon kosher salt
In a large mixing bowl, combine the cooled brown butter and granulated sugar. Use a hand mixer or stand mixer to cream them together until well combined.
1 cup (200 grams) granulated sugar
Add the egg yolk to the butter-sugar mixture and beat until fully incorporated. Then, mix in the sourdough discard, Greek yogurt, and vanilla extract until smooth.
1 large (18 grams) egg yolk, ½ cup (100 grams) sourdough starter discard, ¼ cup (56 grams) plain Greek yogurt, 2 teaspoons vanilla extract
Gradually add the dry ingredients to the wet ingredients, mixing until a thick cookie dough forms.
Cover the cookie dough and refrigerate it for at least one hour or up to four days.
About 20 minutes before the cookie dough is done chilling, preheat the oven to 375°F and line a baking sheet with parchment paper.
Roll heaping tablespoon-sized portions of the chilled cookie dough into balls and place them on the prepared baking sheet, leaving 2 inches of space between each ball. Flatten the cookie dough balls slightly to help them spread. Alternatively, roll the dough out and cut it using a cookie cutter (see notes).
Bake the cookies for 8-9 minutes, or until the tops are crackly and the bottoms are golden brown.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.