The word "carton" in the title refers to a 6-ounce yogurt carton, which is used to measure most of the ingredients in this easy-to-assemble cake. Just make sure to rinse and dry the carton before using it to measure out the dry ingredients.
For the cake: Position an oven rack in the middle of the oven and preheat to 350 degrees F. Spray a 10-inch round cake pan with a layer of nonstick cooking spray.
To a large mixing bowl, add the yogurt, oil and granulated sugar (using the yogurt carton as the measure) and the eggs; beat together with a hand whisk until everything is incorporated and smooth, 1 to 2 minutes.
Next, add the cake flour and cocoa using the carton as the measure, then the vanilla, baking soda and baking powder; beat together until incorporated, 1 to 2 minutes more. Add the milk and mini chocolate chips (using the carton) and mix until incorporated.
Pour the batter into the prepared cake pan and bake until a cake tester inserted into the center of the cake comes out clean, 35 to 40 minutes. Let cool in the cake pan for 30 minutes, then remove the cake to a cooling rack and allow it to cool for another 30 minutes.
For the vanilla buttercream frosting: Add the confectioners' sugar (measured in the carton), butter and vanilla to the bowl of an electric mixer fitted with a paddle attachment and beat until smooth and creamy, 2 to 3 minutes. Spread over the top of the cooled cake in a thick layer and sprinkle with the mini chocolate chips (measure in the carton).