Preheat oven to 350. Line two baking sheets with parchment paper or silicone liners.

Beat butter and sugar for several minutes, until light and fluffy. Add egg, egg white, sour cream and vanilla and beat until thoroughly combined. The mixture might look slightly curdled.

In a separate bowl, combine flour, baking powder and salt. With your mixer on low, add flour to the butter mixture, mixing until just combined.

Scoop the dough into 1 ½ tablespoon balls and place on prepared cookie sheets. Use the palm of your hand to press down the balls of dough. These cookies will not spread much, so make sure you flatten them. Bake cookies for 7-9 minutes, until the tops are just set, but still appear soft. Cool on the baking sheet for 5 minutes, then remove to a rack to cool completely.

Put strawberries, lemon zest and ¼ 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 ¼ 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.

Place the cooled cookies on a rack set over foil or parchment paper (for easy clean up). Use a spoon to dollop the icing into the center of each cookie and slowly swirl it around, allowing it to flow outward toward the edges, and over some of the sides. Repeat with all the cookies. Let the cookies stand at room temperature for at least 2 hours, until the icing firms up.