Using an electric mixer, cream the butter in a large bowl. Gradually add the brown sugar, then the egg, beating until smooth. Blend in the vanilla. Sift the flour, cocoa, baking soda, and salt into another bowl. Using a wooden spoon, stir the dry ingredients into the creamed mixture about half at a time, until uniformly blended. Cover the dough and refrigerate for 30 to 40 minutes.
Preheat the oven to 350°F. Lightly butter two large baking sheets or line them with parchment paper. With lightly floured hands, roll the dough into 1- to 1¼-inch-diameter balls. Place them on the baking sheets, leaving about 3¼ inches in between. Bake one sheet at a time on the center oven rack for 15 minutes. The cookies will spread out into thin disks. If possible, refrigerate the second sheet while baking the first. Cool the cookies on the baking sheet for 2 minutes, then transfer to a rack and cool completely.