Have all ingredients at room temperature Preheat oven to 350 degrees with rack in the lower third of the oven. Spray 8-10 cup tube pan with cooking spray.
Use a whisk to combine four, baking powder, baking soda, and salt. Sift together. Set aside. In a small bowl combine the espresso powder, cocoa and 1/3 cup of the sugar with 1/4 cup of water. Whisk until smooth. Set aside. In another small bowl, whisk whole egg with egg white. Set aside.
Cut the butter into pieces and place in an electric mixing bowl. Beat for 1 minute to soften. Gradually add remaining 1 cup of sugar and beat at high speed for about 3 minutes. Dribble eggs in slowly, about 1 T at a time, beating constantly for about 2 minutes. On low speed, beat in a third of the flour mixture. On medium-high speed, beat in half of the yogurt. On low speed, beat in half of the remaining flour. On high speed, beat in the rest of the yogurt and the vanilla. On low speed, beat in the remaining flour.
Measure out 1 1/2 cups of yogurt batter and mix into the cocoa mixture. Set aside.
Use a large spoon to fill the bottom of the pan with about three quarters of the white batter placed in dollops. Cover the white batter with dollops of chocolate batter. Top the chocolate batter with small dollops of white batter spaced so that the chocolate batter shows through. Use a table knife to marble the batters together with a circular or zigzag motion; be careful not to blend them too much. Bake for 45-50 minutes or until the cake starts to pull away from the sides of the pan and a toothpick inserted in comes out clean.
Cool for 10-15 minutes on a rack. Unmold the cake. Cool completely before serving or storing.