Adjust oven rack to middle position and heat oven to 325°F (163ºC). Line 2 baking sheets with parchment paper.
Melt 3 tablespoons butter in a microwave-safe bowl. Whisk ⅓ cup (40g) cocoa powder and 1 teaspoon espresso power into melted butter until smooth; set aside.
Using a stand mixer fitted with paddle attachment, blend remaining 5 tablespoons softened butter, 1⅓ cups (138g) confectioners' sugar and ¾ teaspoon salt together on low speed, gradually increase speed as it comes together and beat until light and creamy. Scrape down sides of bowl and add cocoa mixture; beat on medium-high speed until fluffy, about 2 minutes, scraping down sides of bowl as needed.
Add egg yolk and 1½ teaspoons vanilla and beat until combined, about 30 seconds.
Decrease mixer to low speed and add 1 cup (130g) flour in three additions, mixing after each addition; scrape down sides of bowl as needed. Continue to mix until a cohesive dough comes together, about 30 seconds.
Using your hands, firmly press dough into a large disk (you may have to work it a bit). Break in half then roll half of dough to ⅛-inch thick between two sheets of parchment paper.
Cut dough into rounds using a 2-inch round cookie cutter. Transfer rounds to prepared baking sheets, spaced 1-inch apart.
Bake cookies 5 minutes, then remove from oven and using a cookie press, lightly press just to imprint design into each dough round (do not press hard or you risk spreading the dough too thin). Return sheet to oven and bake until dry to touch and edges are set, 9–11 minutes more (total bake time 14–16 minutes). Let cookies cool on sheet 5 minutes then transfer to wire racks to cool completely. Repeat rolling and baking with remaining dough and scraps.