for the pudding:

1. Combine the blackberry puree, sugar, and lemon juice into a large saucepan and bring to a simmer. Whisk occasionally as it heats, and remove from the heat once the sugar is dissolved.
2. Combine the cornstarch and salt in a medium saucepan and stir together. Stir in 1/4 cup of the milk and whisk until smooth, ensuring there are no lumps. Whisk in the egg yolks and remaining milk.
3. Whisk about 1/2 cup of the hot puree into the cornstarch mixture to temper the milk mixture, stirring until combined.
4. Pour the tempered cornstarch/berry mixture into the saucepan with the remaining puree, again continuing to whisk vigorously.
5. Return the saucepan to the cooktop and heat over medium heat. Bring the pudding to a boil, whisking frequently.
6. Continue whisking while the pudding bubbles and begins to thicken, about 5 to 6 minutes. Reduce the heat to medium low and, using rubber spatula, stir constantly, for another 10-12 minutes while the pudding continues to thicken.
7. Once it’s significantly thicker, pour the pudding into a shallow, heat-proof container and let sit on the count for a few minutes to cool.
8. Cover with plastic wrap and chill in the fridge until set, about 4 hours.

for the cake: 

1. Preheat oven to 350°F.
2. Lightly coat two 9-inch round cake pans with baking spray. Line bottoms with parchment paper rounds and coat again with baking spray.
3. Cream the butter and sugar in the bowl of an electric mixer fitted with the paddle attachment, until light and fluffy, about 5 minutes. With the mixer on medium speed, add the eggs, 1 at a time, and the lemon zest.
4. Whisk together the flour, baking powder, baking soda, and salt together in medium bowl; set aside.
5. In another bowl, stir together the lemon juice, buttermilk, and vanilla.
6. Add the flour and buttermilk mixtures alternately to the batter, beginning and ending with the flour. Divide the batter evenly between the pans, smooth the tops.
7. Bake for 35-40 minutes, until the tops of the cakes are golden brown and a tester inserted in the center of the cake comes out clean. Transfer cakes to cooling racks and cool 10 minutes in pans. Turn cakes out directly onto racks and remove the parchment paper. Cool completely, about 1 hour.