Preheat oven to 350 degrees and line your muffin tins with muffin/cupcake liners.
In a small stockpot, bring 1 cup of water to a simmer and place a heatproof bowl on top and combine butter with chocolate and cocoa powder. Continue stirring until all is melted and smooth. Remove from heat and set aside to cool until it's warm to the touch.
Whisk flour, baking soda, and baking powder in a small bowl to combine.
In a second bowl, medium sized, whisk the eggs then add sugar, vanilla, and salt until well incorporated. Add the cool chocolate mixture in and whisk well to combine. Put in about 1/3 of the flour mixture and whisk until combined. Add in the sour cream, whisk until all is incorporated then add in the remaining flour mixture and whisk until everything is completely combined and well incorporated.
Divide the batter evenly amongst the muffin cups and bake for 18-20 minutes.
Cool cupcakes in muffin tin until they're cool enough to handle, about 20 minutes, then lift them out and place them on a wire rack to cool completely. Cool to room temperature before icing them.
In the meantime, while cupcakes are cooling, in a bowl of a stand mixer with the paddle attachment, beat the butter to soften it some more. Add in the powdered sugar and vanilla bean paste. Continue beating on low speed until all is incorporated. If your frosting is too thick, add milk or cream, 1 tbsp. at a time. You don't want it too runny.
Once cupcakes have cooled, ice your cupcakes with whatever pastry tip you want.
Store in an airtight container for a week. If you're worried about the frosting going bad, you can keep it in the fridge for up to 2 weeks (although, the cupcake part might get kind of dry and hard).