Turn the mixer onto the slowest speed (setting 1 on Kitchen Aid mixers). Slowly add chunks of your softened butter until it is all added then let everything mix until it looks like coarse sand.
Add your milk/oil mixture all at once to the dry ingredients and mix on medium (speed 4 on KitchenAid) for 1 full minute to develop the structure. Don’t worry, this will not over-mix the cake, because it is AP flour you do not need to mix as long as cake flour (more gluten, less time needed)
Scrape the bowl. This is an important step. If you skip it, you will have hard lumps of flour and unmixed ingredients in your batter. If you do it later, they will not mix in fully.
Slowly add in the rest of your milk/eggs/zest ingredients in 3 parts, pour in the lime juice with the last addition of liquid, stopping to scrape the bowl one more time halfway through. Your batter should be thick and not too runny. Fill your pans 1/2 full with cake batter.
Spoon about 2 tablespoons of room temp puree over the batter, use a small spatula or knife to gently fold the puree in to create a swirl effect into the batter.
Add floured blackberries to the top of the batter, do not push them down in as their weight will cause them to sink anyway as the cake heats up.
I always start by baking for 30 minutes for 6″ and 8″ cakes and 35 minutes for 9″ and larger cakes and then checking for doneness. If the cakes are still really jiggly, I add another 5 minutes. I check every 2 minutes after that until I’m close and then it’s every 1 minute. Cakes are done when a toothpick inserted in the center comes out with a few crumbs.
After cakes have cooled for 10 minutes or the pans are cool enough to touch, flip the cakes over and remove from the pans onto the cooling racks to cool completely. Wrap in plastic wrap and chill in the freezer.
Once the cakes are chilled in the freezer and feel firm you can trim off the brown edges from the sides (optional) and fill with your blackberry buttercream and fresh berries. Crumb coat the cake with buttercream, chill then apply your final coat of buttercream. Top with fresh berries and lime zest.
