Preheat the oven to 350°F and line 14 muffin cups with paper cupcake liners.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until well combined; set aside.
In a large mixing bowl, beat 1/2 cup softened butter, granulated sugar, and lemon zest on medium-high speed until light and fluffy.
Add the eggs one at a time, beating well after each addition and scraping down the bowl as needed, then mix in the vanilla bean paste.
With the mixer on low speed, add half of the sour cream, then half of the flour mixture, mixing just until combined; repeat with the remaining sour cream and flour mixture.
Gently fold in the lemon juice and poppy seeds with a spatula until evenly distributed.
Divide the batter among the prepared liners, filling each about two-thirds full, and smooth the tops if needed.
Bake for 16 to 20 minutes, or until the cupcakes spring back when lightly pressed and a toothpick inserted in the center comes out clean.
Cool the cupcakes in the pan for 10 minutes, then transfer them to a wire rack to cool completely.
While the cupcakes cool, puree the blackberries in a blender or food processor until smooth.
Press the blackberry puree through a fine mesh sieve into a small saucepan to remove the seeds.
Simmer the seedless blackberry puree over low heat for 10 to 20 minutes, stirring occasionally, until thickened to a jam-like consistency; remove from heat and cool completely.
In a large bowl, beat 1 cup softened butter on medium speed until smooth and creamy.
Add 2 cups of powdered sugar and mix on low speed until incorporated, then increase speed and beat until fluffy.
Beat in 2 tablespoons of the cooled blackberry puree, then continue adding the remaining powdered sugar 1/2 cup at a time, alternating with small amounts of additional blackberry puree and the heavy cream, until the frosting is smooth and spreadable.
If needed, adjust the consistency with a few more drops of cream to loosen or a little extra powdered sugar to thicken.
Transfer the blackberry frosting to a piping bag fitted with a large star or round tip, or use an offset spatula.
Pipe or spread generous swirls of frosting onto the cooled cupcakes and serve.