Put strawberries, lemon zest, and 1/4 cup powdered sugar in the bowl of a food processor, fitted with a metal blade. Process until strawberries are completely smooth.
Using a rubber spatula or wooden spoon, press the strawberry puree through a fine-mesh sieve and into a large bowl. Discard the seeds.
Add butter, vanilla, and remaining 3 1/4 cups powdered sugar to the strawberry puree. Whisk until smooth.
If the icing seems too loose, add a little more powdered sugar. If it seems too dry, add a few drops of water. The final consistency should be a little thicker than honey. Depending on the moisture content of your strawberries you might need more or less powdered sugar. You can also refrigerate the icing for 15 minutes to help it firm up, if necessary.
Use on top of your favorite desserts and pastries like Mason Dixie Biscuits or Scones!