Lemon Sugar Cookies

Name: Lemon Sugar Cookies
Ingredients: 270g granulated sugar (9 1/2 ounces; 1 1/4 cups); 6g (2 tablespoons) packed grated lemon zest plus 1 1/2 tablespoons fresh juice from 4 large lemons, divided; 227g (2 sticks) unsalted butter, softened; 1 large egg, room temperature; 5g (1 teaspoon) vanilla extract; 320g all-purpose flour (about 11 1/4 ounces; 2 1/2 cups); 1/2 teaspoon baking powder; 1/2 teaspoon Diamond Crystal kosher salt; for table salt, use half as much by volume; 140g sifted powdered sugar (about 5 ounces; 1 1/2 cups); 3 tablespoons fresh lemon juice from 1 large lemon; 1/4 teaspoon vanilla extract; Lemon zest, for garnish