Preheat oven to 350°F. Butter a 13x9" pan and dust with 2 tbsp cocoa powder, tapping out excess.
Melt remaining 1 cup butter in a small saucepan over medium heat. Add chocolate and remaining ½ cup cocoa powder; cook, whisking constantly, until smooth. Let cool slightly.
Whisk eggs and vanilla in a large bowl. Add sugar and whisk until smooth, lightened in color, and slightly aerated, about 2 minutes. Stream in buttermilk while whisking, then stream in chocolate mixture while whisking until smooth.
Sift flour, baking powder, baking soda, and salt into chocolate mixture. Whisk until batter is mostly smooth (there will be a few small lumps). Transfer to prepared pan.
Bake cake until a tester inserted into the center comes out clean, 35–40 minutes. Transfer pan to a wire rack and let cake cool completely.