For cookies: In the bowl of a stand mixer fitted with the paddle attachment, beat butter and brown sugar at medium speed until light and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating until combined after each addition and stopping to scrape sides of bowl. Beat in vanilla.
In a medium bowl, whisk together flour, cocoa, and salt. With mixer on low speed, gradually add flour mixture to butter mixture, beating until just combined and no dry streaks remain.
Turn out dough onto a clean surface, and gently knead until dough is smooth and combined. Flatten into a disk, and wrap in plastic wrap. Refrigerate until firm, about 1 hour, or up to overnight. (If refrigerating longer than 1 hour, let dough stand at room temperature for about 10 minutes to soften before rolling out.)
Preheat oven to 375°F (190°C). Line rimmed baking sheets with parchment paper.
On a lightly floured surface, roll half of dough to ⅜-inch thickness. (Keep remaining dough covered until ready to use to prevent it from drying out.) Using a well-floured 3-inch cookie press (see Note), firmly press designs into dough as desired. Using a 3-inch round cutter, cut rounds from dough, and place at least 1½ inches apart on prepared pans. Repeat with remaining dough, rerolling scraps to use all dough.
Bake, one pan at a time, until edges are just set, 8 to 10 minutes. Let cool on pan for 10 minutes. Remove from pan, and let cool completely on a wire rack.
For glaze: In another medium bowl, whisk together confectioners’ sugar, 1 tablespoon (15 grams) water, and peppermint extract; add up to remaining 1 tablespoon (15 grams) water until glaze is fluid. Using a pastry brush, brush glaze onto cooled cookies. Let stand until glaze is set. Store in an airtight container for up to 1 week.
