Preheat oven to 350 degrees Fahrenheit and line the bottoms of three 6" cake pans with parchment paper. Butter and flour the pans.

In a medium-sized bowl, whisk together the flour, baking powder, and salt. Set aside.

In a large bowl, using an electric mixer, cream together the butter, oil, and sugar on medium-high speed until light and fluffy (3-4 minutes).

Reduce speed to low and add the eggs one at a time, mixing between each addition. Then, beat briefly on medium speed until the mixture is light, smooth, and creamy. Add the vanilla bean paste and mix to combine.

With the mixer on low speed, alternate adding 1/3 of flour mixture, then 1/2 of the milk, beginning and ending with the flour. Mix between each addition until incorporated. You will have 3 additions of flour and 2 additions of milk.

Divide the batter evenly between prepared cake pans and bake at 350 degrees for 22-25 minutes, or until a toothpick inserted into the cakes comes out with a few moist crumbs attached.

Let the cakes cool in the pans for about 10 minutes, then turn cakes out onto a wire rack to cool completely.

Preheat oven to 375 degrees Fahrenheit and line two rimmed baking sheets with parchment paper.

Gently toss the strawberries, sugar, and balsamic vinegar together in a large bowl. Divide the strawberries evenly between the prepared baking sheets and roast at 375 for 30-35 minutes, until the strawberries have begun to caramelize.***

Place baking sheets on top of a wire rack and allow strawberries to cool completely.

Once cooled, place half (one pan) of the strawberries in a blender or food processor, leaving most of the juices on the pan. Pulse until strawberries are pureed. Set aside for the buttercream and reserve the other pan of strawberries to decorate the top of the cake.

Using an electric mixer, beat the butter on medium-high speed until light and fluffy. With the mixer on low, gradually add the powdered sugar (about 1 cup at a time), mixing to incorporate after each addition. Add salt and vanilla bean paste, then mix to combine.

Add the strawberry puree one tablespoon at a time until desired taste, consistency, and color is reached. I used about 6 tablespoons of puree, from well-roasted strawberries (i.e. they were on the drier side - see notes).

Level each cake layer by using a serrated knife or cake cutter to slice the domes off the top of each cake.

Using a small amount of buttercream, attach the first layer to a six-inch cake board and cover with buttercream. Repeat with the second layer and more frosting. Finally, stack the third layer on top, cut side down.

Frost the top and sides of the cake with a thin layer of buttercream (a crumb coat), then chill the cake for about 20 minutes. Finish frosting as desired, using the remaining buttercream. I used a large, closed star tip to pipe rosettes around the sides of the cake. See directions in the post.

Finally, top the cake with the remaining balsamic roasted strawberries.