Preheat the oven to 350°F (175°C). Grease and flour the 9x13 inch baking pan.
In a small saucepan, combine the fresh blackberries and dried lavender buds with 1/4 cup of water. Cook over medium heat for about 5 minutes until the blackberries break down and release their juices. Strain the mixture through a fine mesh strainer into a bowl, discarding the solids. Allow the blackberry-lavender syrup to cool.
In a mixing bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large mixing bowl, cream the softened butter and granulated sugar using an electric mixer until light and fluffy (about 3-4 minutes).
Add the eggs, one at a time, mixing well after each addition. Then, add in the milk and vanilla extract, mixing until combined.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Divide the batter in half. To one half, add the cooled blackberry-lavender syrup and mix until fully incorporated.
In the prepared baking pan, drop alternating spoonfuls of the vanilla batter and the blackberry-lavender batter. Use a rubber spatula or knife to swirl the batters together gently.
Bake in the preheated oven for 30 minutes or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for about 10 minutes before transferring it to a wire rack to cool completely.
Once cooled, dust the top with powdered sugar before slicing into pieces and serving.