Make the cookies: Pulse the flour, confectioners' sugar, cocoa powder, granulated sugar and salt in a food processor until combined. Add the butter and vanilla and pulse until the mixture looks crumbly and holds together when pinched. Transfer the dough to a large bowl and press into a ball. Knead a few times until the dough comes together. Turn out onto a piece of parchment paper. Top with another piece of parchment and roll out the dough until 3/8 inch thick. Refrigerate until firm, about 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 325 degrees F. Cut out cookies using a 3-inch round cookie cutter. Arrange 2 inches apart on 2 unlined baking sheets, then insert an 8-inch lollipop stick about three-quarters of the way into each. Gather the scraps and refrigerate until firm, if needed; reroll to cut out more cookies.
Bake, switching the pans halfway through, until the cookies are set, 25 to 30 minutes. Let cool 10 minutes on the pans, then transfer to a rack to cool completely.
Meanwhile, make the icing: Whisk the confectioners' sugar and meringue powder in a large bowl. Beat in the water and cherry extract with a mixer on medium speed until soft glossy peaks form, about 3 minutes; add up to 1 more tablespoon water as needed until the icing is pipeable but not runny. Tint red with food coloring.
Decorate the cookies: Transfer about 1/2 cup icing to a resealable plastic bag and snip a corner. Pipe the icing around the edge of each cookie; set the bag of icing aside for adding details later. Thin the icing in the bowl until the consistency of runny yogurt, adding 1 teaspoon water at a time. Transfer to another resealable plastic bag and snip a corner; pipe within the outline, then use the back of a spoon or an offset spatula to smooth it out. Let set, about 1 hour. Pipe the reserved detail icing on top of the cookies to make a floral design.