Sift together flour, cocoa, salt, and cinnamon. With an electric mixer on medium-high speed, cream butter and confectioners' sugar until pale and fluffy. Add egg and vanilla, and beat to combine. Reduce speed to low. Gradually add flour mixture, beating until just combined.
Divide dough in half; flatten each half into a disk, and wrap in plastic. Refrigerate until firm, at least 1 hour (or overnight). Dough can be frozen, wrapped tightly in plastic and placed in a resealable plastic bag, up to 3 months; thaw overnight in the refrigerator before using.
On a lightly floured work surface (or a piece of parchment paper), roll out one disk of dough to just under 1/4 inch thick. Transfer to a baking sheet; freeze until firm, about 15 minutes.
Preheat oven to 350 degrees, with racks in upper and lower thirds. Using a 3-inch cookie cutter, quickly cut out shapes from dough (if dough begins to soften, chill in freezer 5 minutes), and transfer shapes to parchment-lined baking sheets, spacing them 2 inches apart, as you work. Reroll scraps and cut out more shapes. Brush off excess flour. Freeze until firm, about 15 minutes. Repeat with remaining disk of dough.
Bake cookies until crisp, about 8 minutes, firmly tapping down sheets once and rotating them halfway through. Let cool completely on sheets on wire racks before decorating as desired.