Preheat oven to 350 degrees. Add the butter, water, and cocoa to a medium microwave safe bowl. Cook for about 2-2 1/2 minutes whisking 30 seconds at a time until melted and smooth. Set aside.
In a medium sized bowl combine eggs, yogurt and applesauce and whisk until smooth. Set aside.
In a large mixing bowl, combine flour, sugar, baking soda, and salt. Whisk together. Add the warm chocolate mixture and stir to combine. Lastly add the yogurt mixture and stir again until combined. The batter may have a few lumps, but that is ok.
Pour into an ungreased jelly roll pan (13x18) or two 9x13 baking sheets and bake for 15-20 minutes until toothpick inserted in the center comes out clean.
To make the frosting:
Melt the butter in a microwave safe bowl. Add cocoa and whisk until smooth. Whisk in milk and vanilla and lastly whisk in the powdered sugar. Pour over the cake and frost to the edges. Let the cake cool completely and allow the frosting to set.