Preheat oven to 350°F and spray two 8-inch round cake pans with nonstick spray. Whisk together flour, cocoa, baking powder, baking soda, and salt in a large bowl; make a well in the center of the mixture. Add sugars, eggs, egg yolk, milk, yogurt, oil, and vanilla to flour mixture; whisk in just until incorporated. Add boiling water to batter and whisk in just until smooth. Fold in chopped chocolate.
Divide batter evenly between cake pans and smooth with an offset spatula. Tap pans sharply to reduce air bubbles.
Bake for 28 to 30 minutes, until cakes spring back to the touch, a toothpick inserted in the center comes out with a few crumbs, and the edges are pulling slightly away from the pan. Cool cakes completely in pans on a wire rack.