1/2 pound (2 sticks) unsalted butter, at room temperature
2  1/2 cups granulated white sugar, divided
4 extra large eggs, at room temperature
1/3 cup freshly grated lemon zest
3 cups all-purpose flour
1/2 teaspoon baking powder
1/2 teaspoon baking soda
1 teaspoon kosher salt
3/4 cup freshly squeezed lemon juice, divided
3/4 cup buttermilk, at room temperature
1 teaspoon vanilla extract
2 cups powdered sugar, sifted
3  1/2 tablespoons freshly squeezed lemon juice