Preheat your oven to 350°F (175°C). Grease the 9x13 inch baking pan and line the bottom with parchment paper for easy removal.
In a small saucepan, combine the blackberries, dried lavender, and lemon juice. Cook over medium heat for about 5 minutes, stirring until the blackberries break down and the mixture thickens. Set aside to cool.
In a large mixing bowl, cream together the softened butter and granulated sugar with an electric mixer until light and fluffy.
Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking powder, and salt.
Gradually add the dry ingredients to the butter mixture, alternating with the whole milk. Mix until just combined.
Divide the cake batter in half; gently fold the blackberry-lavender mixture into one half of the batter until fully combined.
In the baking pan, alternate spoonfuls of the plain batter and the blackberry-lavender batter. Using a knife or skewer, gently swirl the batters together to create a marbled effect.
Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes before transferring to a wire rack to cool completely.