Preheat your oven to 350°F (175°C). Grease the 9x13 inch baking sheet or line with parchment paper.
In a mixing bowl, sift together the flour, baking powder, and salt.
In another bowl, cream the softened butter and granulated sugar until light and fluffy using an electric mixer or a whisk.
Add the eggs one at a time, mixing well after each addition. Then, mix in the vanilla extract.
Gradually add the dry ingredients to the wet mixture, alternating with the milk. Start and end with the dry ingredients. Mix until just combined.
In a saucepan, combine fresh blackberries, sugar, lemon juice, water, and culinary lavender. Cook over medium heat until the blackberries burst and the mixture thickens slightly, about 5-7 minutes. Let cool slightly.
Once cooled, gently fold half of the blackberry mixture into the cake batter to create a swirl effect.
Pour the batter into the prepared baking sheet. Drop spoonfuls of the remaining blackberry mixture on top of the batter and swirl it gently with a knife or a toothpick.
Bake in the preheated oven for 25-30 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool in the pan for 10 minutes before transferring it to a cooling rack to cool completely.
If desired, prepare the frosting by mixing the powdered sugar, softened butter, milk, and vanilla extract until smooth. Adjust the milk for your desired consistency.
Frost the cooled cake if desired and serve.