Use a hand or stand mixer at medium speed to blend the butter and sugar. Beat in the egg, then stir in the vanilla extract. Slowly add the flour and cocoa powder and mix until a soft dough forms.
Halve the dough and flatten each portion into a large disk. Cover each disk with plastic wrap and chill them in the refrigerator for 1 hour.
Heat the oven to 375°. Unwrap one portion of dough. Sandwich it between two sheets of plastic wrap, then roll it out to a 1/8-inch thickness. Use a circular cookie cutter or the lid of a small spice jar to cut the dough into small rounds (our cutter was 1 3/4 inches). Place the cookies on an ungreased baking sheet. Use the rounded edge of a slightly smaller cutter or the rim of the spice jar to make an indentation around the perimeter of each cookie. Poke four button holes in each cookie with the flat tip of a bamboo skewer.
Bake the cookies for 6 minutes. Transfer them to a rack to cool completely. Repeat the rolling, shaping, and baking steps with the remaining dough portion and scraps. To pipe stitches onto each cookie using Vanilla Decorating Icing.
In a small bowl, whisk together the sugar, milk, corn syrup, and flavored extract until smooth.
The dough can be made up to two days ahead. If it's been chilled for several hours, Let it sit at room temperature for 15 minutes before proceeding to the next step.