Strawberry-Filled Cake - The Local Palate

Ingredients

• 3 cups flour
• 1 teaspoon salt
• 3 teaspoons baking powder
• 1 cup butter, at room temperature
• 2 cups sugar
• 4 eggs, at room temperature
• 2 teaspoons vanilla
• 1 cup milk
• 1 pound fresh strawberries, cut into ¼-inch pieces
• ½ cup sugar, or to taste (too much is too much!)
• ½ teaspoon vanilla
• ½ teaspoon fresh lemon juice
• 1 stick unsalted butter, room temperature
• 5 ounces strawberry puree (from approximately 1 pound strawberries)
• 8 cups confectionary sugar, sifted
• 1-2 tablespoons heavy cream, plus more if needed