Preheat the oven to 350°F Grease a 9-inch square metal baking pan and line the bottom with parchment paper; grease the parchment.
In a large bowl, combine the flour, cocoa, baking powder, espresso powder and salt. Using an electric mixer, beat the sugar, eggs and egg whites at medium-hight speed until pale in color, about 3 minutes; whisk in the olive oil. Stir in the zucchini. Stir in the flour mixture and 1 container of yogurt alternately in 2 batches. Transfer to the prepared pan.
Bake until springy to the touch and a cake tester comes out with moist crumbs, about 35 minutes; let cool slightly. Serve warm with the remaining yogurt.