Preheat the oven to 350°F. Line 2 cookie sheets with parchment paper.
To make the dough: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, powdered sugar, and vanilla extract. Mix on low until combined and you don’t see powdered sugar anymore, there will still be a few chunks of butter but that is okay because it’s very soft.
Add the flour and freeze dried strawberries into the butter mixture and mix on low until combined and smooth. The dough should feel like play dough in your hands.
Using your hands, form the dough into 24 balls and place them on each prepared baking sheet, you'll have 12 cookies per sheet. Flatten each one slightly so they are about 2-inches in diameter.
Bake for 18 to 21 minutes or until golden brown on the bottoms and edges. Sprinkle with cane sugar immediately after coming out of the oven. Allow to cool completely on the baking sheets.
To make the filling: In the bowl of a stand mixer fitted with the paddle attachment, add the powdered sugar, butter, zest of half a lemon, and the juice of half a lemon, and mix on low until combined. Speed mixer up to high for 1 minute or until light and fluffy. Put in a piping bag with Ateco tip #864.
Turn over every other cookie so they are all paired. Pipe a swirl of filling onto each cookie and sandwich them together.
Store in a cool dry place, or in an airtight container for up to 7 days.