Chocolate Frosted Chocolate Sheet Cake

Cake
1/2 cup (230 g) salted butter
1/4 cup (25 g) cocoa
1 cup (240 ml) water
1/2 cup (240 ml) yogurt, sour cream, or buttermilk
1 teaspoon (5 ml) vanilla extract
2 cups (240 g) all-purpose flour OR 1 cup (120 g) all-purpose flour plus 1 cup (100 g) oat flour
2 cups (400 g) granulated sugar
1 teaspoon (4 grams) baking soda
2 large eggs, slightly beaten
Frosting
1/4 cube (115 g) salted butter
1/4 cup (25 g) cocoa
6 tablespoons (90 ml) milk
1 teaspoon (5 ml) vanilla extract
1 pound (455 g or 3 3/4 cups) powdered sugar

Heat the oven to 350º F.
Place the butter, cocoa, and water in a small saucepan over medium low heat. Heat until the butter is melted, stirring frequently.
When the butter is melted, whisk in the yogurt and vanilla.
In a mixing bowl, combine the flour, sugar, and baking soda.
Pour the liquid mixture into the dry mixture all at once. Mix until combined, about 1 minute. Scrape the sides of the bowl.
Add the eggs and mix to completely combine, about 1 minute. Scrape the sides of the bowl and mix for 30 more seconds.
Pour the cake into a 16 x 11″ pan (quarter sheet pan). You can also use a 13 x 9 x 2″ pan.
Bake until a toothpick comes out just clean. For a 1/4 sheet pan, this should be about 20 minutes. For a 13 x 9 x 2 pan, this should be about 25-30 minutes.
While the cake is cooking, make the frosting. Place the butter, cocoa, and milk in a small saucepan over medium low heat. Heat until the butter is melted, stirring frequently.
Pour the hot chocolate mixture into a mixing bowl with the powdered sugar. Mix on low until well-blended, about 1 minute. Scrape side of bowl occasionally.
Pour the frosting over the warm cake. Allow the cake to cook completely.