Increase the speed to medium-high and gradually beat in the sugar mixture, scraping down the bowl as needed. Continue beating until the sugar is fully dissolved, 3 to 4 minutes. (You can test this by rubbing a small amount of frosting between your fingers, and if it feels grainy, keep beating.) Add the lemon juice, 1/2 teaspoon vanilla extract, and a pinch of kosher salt, and beat on medium speed until incorporated. Refrigerate for at least 30 minutes.
