Preheat the oven to 350°F. Position the oven rack in the bottom third and generously grease a Bundt or large round cake pan with butter.
Combine softened butter, granulated sugar, and salt in a large mixing bowl. Using an electric hand mixer or a stand mixer with a paddle attachment, beat on medium speed for 2 to 3 minutes until pale, smooth, and creamy.
Scrape down the sides of the bowl with a silicone spatula, then add the vanilla and mix for 30 seconds.
Add eggs one at a time, beating for 30 seconds and scraping the bowl after each addition. Continue until all eggs are incorporated, even if the batter appears curdled.
Pour in the milk and beat for an additional 30 seconds, then set the mixture aside.
Whisk the flour and baking powder together in a separate large mixing bowl.
Gradually add the flour mixture to the egg mixture while mixing on low speed. Scrape the bowl to ensure no dry pockets remain in the thick batter.
Transfer two-thirds of the cake batter into the prepared baking pan.
Add cocoa powder, espresso powder, and milk to the remaining batter. Mix until fully combined, scraping the sides of the bowl as needed.
Pour the chocolate batter over the light batter. Drag a knife through the mixtures to create a marbled effect, taking care not to over-mix and lose the contrast.
Bake for 45 to 60 minutes until a toothpick or knife inserted into the center comes out clean.
Cool the cake on a rack for 10 minutes, then invert the pan to release it. Allow the cake to cool completely on the rack before dusting with powdered sugar.
