Whisk egg white in a medium bowl just to loosen. Add powdered sugar and 1/2 tsp. water and whisk until smooth. Mixture should be smooth, satiny, and very thick. If mixture is too thick or gloppy, whisk in another 1/2 tsp. water. Transfer to a disposable pastry bag and snip off end of bag (or use a resealable plastic bag and cut off 1 corner) to make a very small opening.
Pipe icing in very thin, very straight lines across a parchment-lined baking sheet, spacing close together (if icing does not squeeze out of bag easily, snip a slightly larger opening). Keep piping until you’ve covered the entire sheet of parchment (discard leftover icing or save for another use). Let sit at room temperature until hardened, at least 8 hours and up to 1 day (to speed up drying, you can also bake sprinkles in a 150°F oven for 2–2 1/2 hours). Lift ends of parchment to break sprinkles into various lengths, then gently roll your fingers over top to loosen from parchment paper.
Preheat oven to 350°F. Heat chocolate and butter in a heatproof bowl set over a saucepan of barely simmering water (do not let water touch bottom of bowl), stirring until melted and mixture is smooth. Let cool.
Meanwhile, sift flour, cocoa, baking powder, espresso powder, and baking soda in a medium bowl. Using an electric mixer on medium-high speed, beat eggs, brown sugar, vanilla, and salt in a large bowl until thick and light, about 3 minutes. Reduce speed to low and add chocolate mixture; mix just to combine. Turn off mixer. Add dry ingredients and pulse mixer on low until no dry spots remain. If dough is very soft, cover and chill until firm enough to scoop, 10–15 minutes.
Using a 1 1/2 oz. scoop (or a scant 1/4 cup), portion out dough and roll into balls between your palms (if dough becomes very sticky while rolling, chill for a few minutes). Place balls on 2 parchment-lined baking sheets, spacing at least 2″ apart; flatten slightly. Bake cookies, rotating baking sheets once, until centers are puffed and edges are just firm,13–18 minutes. Let cookies cool 20 minutes on baking sheets, then peel away parchment and transfer to a wire rack. Let cool completely.
Icing and assembly:
Heat chocolate, cream, corn syrup, and salt in a heatproof bowl set over a saucepan of barely simmering water (do not let water touch bottom of bowl), stirring until chocolate is melted and mixture is smooth. Let cool.
Whisk powdered sugar and vanilla into chocolate mixture, then add butter 1 piece at a time, whisking to incorporate fully after each addition. Whisk until icing looks smooth and glossy.
Using an offset spatula, spread about 1 Tbsp. icing over each cookie and top with sprinkles.