Creating the Swirled Blackberry Lavender Sheet Cake is simple if you follow these steps:

1. Preheat the Oven: Preheat your oven to 350°F (175°C). Prepare a 9×13-inch baking sheet by greasing and lining it with parchment paper.

2. Mix Dry Ingredients: In a large bowl, whisk together the flour, baking powder, baking soda, and salt.

3. Cream Butter and Sugar: In another bowl, beat the softened butter and granulated sugar until fluffy, about 3-4 minutes.

4. Incorporate Eggs and Vanilla: Add the eggs, one at a time, mixing well after each addition. Stir in the vanilla extract.

5. Combine with Dry Ingredients: Gradually add the dry mixture to the butter mix, alternating with the buttermilk. Start and end with the flour mixture, and blend until just combined.

6. Prepare Blackberry Lavender Mixture: In a small saucepan, heat the blackberries and lavender over low heat until the blackberries break down into a sauce. Let it cool slightly.

7. Swirl the Mixture: Take about 1 cup of the cake batter and mix it with the blackberry-lavender sauce. In the baking sheet, pour the regular batter first and then dollop the blackberry mixture on top. Use a knife or toothpick to swirl it gently.

8. Bake the Cake: Place the baking sheet in the preheated oven and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.

9. Cool the Cake: Once baked, remove from the oven and let the cake cool in the pan for about 20 minutes. Then, transfer it to a wire rack to cool completely.

10. Make the Frosting: In a bowl, whip together the cream cheese, powdered sugar, and vanilla extract. Add milk gradually until you reach your desired frosting consistency.

11. Frost the Cake: Once the cake is fully cooled, spread the frosting evenly over the top. Decorate with additional blackberries and sprigs of lavender if desired.