Preheat the oven to 180°C (356°F), fan mode.
In a stand mixer, beat the butter and powdered sugar on medium-high speed until light and fluffy.
Gradually add the eggs one at a time, beating well after each addition until fully incorporated.
Mix in the yogurt, vanilla extract, and cooled espresso coffee, continuing to beat until smooth. The mixture may look slightly curdled at this stage; it will come together once the dry ingredients are added.
Sift together the flour, baking powder, and cocoa powder before folding them into the batter. Fold gently until the batter is fully combined.
Pour the batter into 20 × 30 cm (8 × 12-inch) baking pan lined with parchment paper, spreading evenly.
Bake for 30 minutes until a toothpick inserted into the center comes out clean. While still warm, evenly drizzle or brush the remaining espresso.Allow the cake to cool completely in the pan before frosting. In a bowl, beat the cream cheese, powdered sugar, and vanilla until smooth and creamy. Spread the frosting evenly over the cooled cake.
Slice into portions and serve.