In a stand mixer fitted with a paddle attachment, cream the butter, granulated sugar, powdered sugar, and salt until thoroughly combined, about 3 minutes.
1 cup unsalted butter, ¼ cup granulated sugar, ¼ cup powdered sugar, ¼ teaspoon kosher salt
In a separate bowl, sift together the all-purpose flour, cake flour, and finely ground freeze-dried strawberries.
1 cup all-purpose flour, 1 cup cake flour, ½ cup freeze dried strawberries With the mixer on low, mix the flour mixture into the butter mixture in 3 additions, scraping down the sides of the bowl in between each addition.
Tear off a large piece of plastic wrap and place dough in the center.
Shape the dough into a smooth, round disk and wrap tightly in the plastic wrap.
Refrigerate for at least 4 hours or overnight.
Remove the dough from the fridge and allow to warm up slightly for about 10-15 minutes.
Preheat the oven to 325ºF.
With a rolling pin lightly floured, roll dough out to ¼ inch thickness.
Using a cookie cutter, cut out dough and place on a parchment-lined baking sheet, spacing cookies 1 inch apart (cookies won’t spread much at all).
Bake the cookies until just barely golden at the edges, about 10-11 minutes.
Remove from the oven and allow to cool completely on baking trays.