Salted Chocolate Chunk Shortbread Cookies

Egg, large

All-purpose flour

Brown sugar, light

Dark chocolate, semi-sweet or bittersweet

Demerara sugar

Granulated sugar

Sea salt, Flaky

Vanilla extract

1 cup plus 2 tablespoons/255 grams total (2¼ sticks) salted butter, cold (room temperature if you're using a handheld mixer), cut into ½-inch pieces (see note)