Adjust oven rack to lower-middle position; heat oven to 350°F. Line a standard-sized muffin pan (cups have 1/2-cup capacity) with baking-cup liners.
Combine butter, chocolate, and cocoa in medium heatproof bowl. Set bowl over saucepan containing barely simmering water; heat mixture until butter and chocolate are melted and whisk until smooth and combined. Set aside to cool until just warm to the touch.
Whisk flour, baking soda, and baking powder in small bowl to combine.
Whisk eggs in second medium bowl to combine; add sugar, vanilla, and salt and whisk until fully incorporated. Add cooled chocolate mixture and whisk until combined. Sift about one-third of flour mixture over chocolate mixture and whisk until combined; whisk in sour cream until combined, then sift remaining flour mixture over and whisk until batter is homogenous and thick.
Divide batter evenly among muffin pan cups. Bake until toothpick inserted into center of cupcakes comes out clean, 18 to 20 minutes.
Cool cupcakes in muffin pan on wire rack until cool enough to handle, about 15 minutes. Carefully lift each cupcake from muffin pan and set on wire rack. Cool to room temperature before icing, about 30 minutes.
Store leftover cupcakes (frosted or unfrosted) in the refrigerator, but let them come to room temperature before serving.