Mix flour, cocoa powder, and salt in a small bowl. Set aside.
Combine butter and sugar in the bowl of a standing mixture on medium speed with the paddle attachment. Beat until light and fluffy. Add egg and vanilla extract and beat until well combined. Add flour mixture and stir on low speed until the dough comes together, about 1 minute. Turn the dough out onto a lightly floured surface and knead 2 or 3 times. Flatten into a disk, wrap in plastic, and chill until firm, about 30 to 40 minutes.
Preheat oven to 350° F. Lightly grease 24 standard-size muffin tin wells.
Roll out the chilled dough to about 1/8-inch thickness on a counter generously dusted with cocoa powder.
Using a 3 3/4-to 4-inch flower-shaped cookie cutter, stamp out as many flowers as possible, rerolling the scraps as necessary to make 24 flowers. Gently transfer the dough to the greased muffin tins, tucking each flower carefully into one of the cups.
With a toothpick or sharp skewer, prick holes in the dough around the edge of the cup, about 1/8 to 1/4 inch apart, adding a few holes in the center as well. Gather the scraps, re roll, and stamp out additional cookies to fill the remaining cups.
Bake in the preheated oven for about 10 minutes, until just set, shifting the baking sheets halfway through for even baking.
Let the cookies cool in the muffin tins for 5 to 10 minutes before running a thin metal spatula around the outside edge of each cup. Leave cups in the pan to cool completely before removing carefully.