Preheat oven to 350°. Grease a 9x9 square baking pan.
Mix together dry flour, baking powder, baking soda, and salt in a small bowl.
Add softened butter and sugar to a medium bowl. Using an electric mixer or a whisk, cream the butter and sugar until completely combined.
Add the eggs one at a time, mixing after each addition. Add the buttermilk and flour mixture in three parts, mixing after each addition. Add vanilla and lemon zest with the final buttermilk addition.
Scrape the batter into the prepared pan and use a spatula to make sure it’s distributed evenly. Bake for 12 minutes, or until a toothpick inserted into the center comes out clean. Let cool at least 15 minutes in the pan. The cake will pull away from the sides as it cools.
Add the mascarpone, sugar, vanilla, lemon juice, and milk to a bowl. Using an electric hand beater or stand mixer with the whisk attachment, beat all ingredients together until smooth and fluffy, about 2 minutes. Scrape into a pastry bag or the corner of a gallon ziploc with the tip cut off.
Cut the cake in half to create two rectangles and trim the edges so all sides are exposed, just like how you would cut the crusts off of toast.
Place the first layer on your serving platter. Using the pastry bag, pipe dots or a ruffled line around the perimeter of the cake to create a decorative edge. Fill in the center with a thin layer of the filling. Place rows of blackberries across the short way, leaving a little space in between each row to make it easier to cut slices later.
Place the second layer on top and, following the same steps for decorating, pipe a decorative perimeter and place rows of blackberries. Pipe dots in between the blackberries to fill in the top.
Refrigerate if not serving right away. This cake should hold up well in the fridge, although I can only attest to one day, as that’s how long it lasted in my house.