Preheat oven to 375°.
Combine sugar, cocoa, flour, and baking powder in a large bowl and whisk until combined.
Add the chopped chocolate and boiling water, stirring until chocolate melts.
Stir in egg yolks and vanilla.
Beat egg whites at high speed with an electric mixer until foamy.
Add ½ cup granulated sugar, 1 Tablespoon at a time, beating until stiff peaks form and sugar dissolves, about 3 minutes.
Stir ⅓ of egg whites into chocolate mixture, then fold in the remaining egg whites.
Pour into a lightly buttered 8" springform pan. Bake at 375° for 28 minutes or until a wooden pick inserted in center comes out clean.
Cool in pan for 10 minutes, then remove sides of pan and cool completely. The cake will crack on top. Sprinkle with powdered sugar if desired.