Lemon Cake With Lemon Poppy Seed Buttercream

Name: Lemon Cake With Lemon Poppy Seed Buttercream
Ingredients: 1 cup (2 sticks) unsalted butter, at room temperature; 1 cup granulated sugar; zest of 1 large lemon; 4 large eggs, at room temperature; 1 vanilla bean, seeds scraped and pod discarded (or 2 teaspoons pure vanilla extract); 2 cups all-purpose flour; 2 teaspoons baking powder; 1/4 teaspoon salt; 1/4 cup fresh lemon juice; 1/2 cup (1 stick) unsalted butter, at room temperature; 2 to 3 cups powdered sugar; 2 tablespoons fresh lemon juice; 1 teaspoon pure vanilla extract; splash of milk or cream as needed for consistency; 1 tablespoon poppy seeds