Preheat the oven to 350°F (176°C) and position a rack in the center of the oven. Butter and flour a 10- to 12-cup Bundt pan or fluted tube pan and tap out the excess flour.
Combine the brewed coffee and espresso powder in a glass measuring cup and let cool.
Sift the flour, baking powder, and salt together in a bowl.
Beat the butter in the bowl of a heavy-duty stand mixer fitted with the paddle attachment on high speed until smooth, about 1 minute. Gradually beat in the sugar, then add the vanilla and beat, scraping the sides of the bowl occasionally, until the mixture is very light in color and texture, about 4 minutes. Beat in the egg yolks, 1 at a time.
Reduce the mixer speed to low. Add the flour mixture in 3 additions, alternating with the cooled coffee in 2 additions, beginning and ending with the flour mixture. Beat until smooth after each addition.
Beat the egg whites in a bowl with a handheld electric mixer on high speed or a balloon whisk until soft peaks form. Using a spatula, stir about 1/4 of the egg whites into the batter, then gently fold in the remaining whites. Scrape the batter into the prepared pan. Smooth the top with the spatula.
Bake until the top of the cake springs back when gently pressed with your finger and a cake tester inserted into the center of the cake comes out clean, 45 to 60 minutes.
Cool the cake on a wire rack for 10 minutes. Invert and unmold the cake onto the wire rack set over a baking sheet. Cool completely.