Line a fine-mesh strainer with cheesecloth or a clean kitchen towel and set it over a bowl. Pour the plain yogurt into the lined strainer, then cover and refrigerate for at least 6 hours, preferably overnight. This step removes excess liquid (whey) from the yogurt, concentrating it into a thick, cream cheese-like consistency that's essential for a frosting that won't slide off your cake. I always use Fage Total 5% because it's already thicker than most yogurts, so even 6 hours of straining gives me the perfect texture.
While the yogurt is draining, sift the powdered sugar into a small bowl to break up any lumps. Sifting ensures a smooth frosting without grittiness and takes just a minute. Set the sifted sugar aside until you're ready to mix.
Once the yogurt has finished straining, transfer the thick strained yogurt to a clean mixing bowl. Add the sifted powdered sugar from Step 2, vanilla extract, and lemon zest. Using an electric mixer on medium-high speed, beat the mixture for about 2 minutes until the frosting is light, fluffy, and holds soft peaks. The lemon zest brightens the flavor without adding liquid, and I find that beating it just long enough to incorporate air gives the frosting a beautiful, cloud-like texture that spreads or pipes beautifully.
Immediately spread or pipe the frosting onto cooled cake layers, cupcakes, or whatever you're decorating. The frosting is best used right after mixing while it's at its fluffiest. If it sits at room temperature for too long, it will begin to soften and lose some of its structure.