To prepare the cupcakes, preheat the oven to 350°F. Line four 6-cup muffin pans with paper liners. 

In a bowl, sift together the flour, baking soda, salt and cocoa powder; set aside. 

In the bowl of an electric mixer fitted with the paddle attachment, beat the butter until light and fluffy. Add the granulated sugar and vanilla and beat until combined. Add the eggs one at a time, beating after each addition until smooth.

On the lowest speed, add the flour mixture in three additions, alternating with the milk and beginning and ending with the flour. Beat just until smooth and fully combined; don’t overwork the batter.

Scoop the batter into the prepared muffin cups, filling each about two-thirds full (don’t bother to smooth the tops, as the batter will level itself as it bakes). 

Bake until the tops of the cupcakes are puffed and spring back when lightly pressed, about 25 minutes. Be careful not to overbake them, but keep in mind that if you take them out too early, they may sink a bit. 

Let the cupcakes cool in the pans for about 5 minutes, then transfer the cupcakes to a wire rack and let cool completely.

To prepare the blackberry-cream cheese frosting, in the bowl of an electric mixer fitted with the paddle attachment, beat together the cream cheese and butter until light and fluffy. Add the blackberry puree and lemon zest and beat until well combined.

Beat in the confectioners' sugar 1/2 cup at a time until the desired sweetness and consistency are reached. 

Spread the frosting over the cooled cupcakes and serve.