Arrange rack in center of oven and preheat oven to 350 degrees F. Spray a 9 by 13-inch cake pan with cooking spray and set aside.
Whisk together flours, cocoa, salt, baking soda and baking powder in a medium bowl, then sift ingredients through a fine mesh strainer.
In a large mixing bowl, whisk together melted butter and oil. Add eggs and egg whites and whisk to incorporate. Fold in yogurt, vanilla, sugar and dissolved espresso powder. Melt the chocolate in the microwave for 90 seconds or over a double boiler. Fold chocolate into batter.
Gradually add sifted dry ingredients and stir until just incorporated; do not overbeat. Pour batter into prepared pan. Bake for 25 to 30 minutes, until cake has risen nicely and a toothpick inserted into the center comes out clean. Cool completely.
While cake is cooling, make the frosting:
Combine all ingredients in a bowl and beat with an electric hand mixer until soft and creamy. Spread frosting evenly over cake and cut into squares. Finely grate one small square of chocolate. Sprinkle the chocolate shavings over the cake.