Preheat oven to 350°. Line a square baking pan (9x9 inch) with parchment paper and lightly spray with baking spray. Set aside.
Melt butter and sugar together, in a large saucepan over medium heat, until sugar dissolves. Whisk in cocoa. Let the mixture cool slightly.
Whisk together, in a small mixing bowl, the ingredients for the fake egg (2 tablespoons of water + 2 teaspoon baking powder + 1 teaspoon oil). Add buttermilk, applesauce and vanilla; whisk to combine.
Add the buttermilk mixture to the cocoa mixture. Add flour; whisk to combine. Transfer batter to prepared pan.
Bake brownies for 35 - 40 minutes, or until set and a toothpick inserted in center comes out with just a few moist crumbs. Cool completely before frosting and cutting.
Make the Frosting: beat all the ingredients with an electric mixer until smooth and fluffy.
Remove brownies from pan. Spread frosting over top of brownies. Garnish with white chocolate curls, if desired (see notes). Cut into 9 large squares and serve.