Lemon cake with lemon buttercream - chinese grandma

Ingredients

• 3 cups cake flour (or 2 3/4 cups all purpose flour)
• 4 teaspoons baking powder
• 1/2 teaspoon salt
• 1-2 tablespoons lemon zest
• 1/2 cup (1 stick) butter, softened
• 2 cups sugar
• 4 eggs, room temperature
• 1 cup milk
• 1/2 cup applesauce
• 1 tablespoon lemon juice
• 1 teaspoon vanilla extract
• 1/2 cup (1 stick) butter, softened at room temperature (don’t use microwave – if you are in a hurry, just cut butter in to cubes to speed softening)
• 1/4 teaspoon salt (IF using unsalted butter)
• 3 cups powdered sugar
• 2 tablespoons lemon juice
• 1 tablespoon milk (or another tablespoon lemon juice for a more lemony frosting)