Preheat the oven to 350°F. Line an 8x8-inch square baking pan with parchment paper, or grease with non-stick cooking spray.
In a large mixing bowl, using a hand whisk, mix together the dry ingredients: flour, sugar, cocoa powder, baking powder, baking soda, and salt.
Add egg, milk, vegetable oil, and vanilla extract. Using an electric mixer, beat on medium speed for about 1-2 minutes or until all ingredients are well combined. Don’t forget to scrape the bottom and the sides of the bowl to make sure ingredients are well incorporated.
Pour boiling water into the batter. Beat on low speed until well combined. You will have a thin batter. Pour the chocolate batter into the prepared baking pan. Spread evenly.
Bake in the preheated oven for about 30-35 minutes or until a toothpick inserted in the center comes out clean.
Remove the pan from the oven and place on a cooling rack for about 10-15 minutes. Then, remove the cake from the pan and let cool to room temperature.
When the cake is cooled, prepare the chocolate frosting. In a saucepan, melt butter. Whisk in cocoa powder for about 1-2 minutes or until fully combined. Remove the saucepan from the heat. Add milk, powdered sugar, vanilla extract, and salt. Beat, using an electric mixer for a few minutes or until smooth and slightly thickened. Spread the chocolate frosting over the cooled chocolate cake. Slice into 9 or 16 squares and serve!