Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy, about 2-3 minutes.
Add the egg, vanilla extract, and lemon zest to the butter mixture, mixing until well combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the crushed freeze-dried strawberries until evenly distributed throughout the dough.
Scoop tablespoon-sized portions of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes, or until the edges are lightly golden. Remove from the oven and let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Once cooled, dust the cookies with powdered sugar before serving.