In a small food processor, combine about 1/3 of the flour with the freeze-dried strawberries. Pulse until the strawberries are ground to a fine powder. Pour this mixture into a medium bowl with the remaining flour and whisk to combine.

If you don't have a food processor, place the freeze-dried strawberries in a zip-top bag and crush them with a rolling pin until powdery, then whisk into the flour.

In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, sugar, vanilla, and salt on medium speed until smooth and creamy, about 2 minutes. Scrape down the sides and the paddle.

Add the flour-strawberry mixture all at once. Mix on low speed until the dough comes together in a cohesive ball. It will be crumbly at first but will form a ball.

Turn the dough out onto a lightly floured surface (or a Silpat mat). Shape into a rectangle, lightly flour the top, and roll to 1/2-inch thickness.

Using a 2-inch round cutter, cut out cookies and place them on a parchment-lined baking sheet. Prick each cookie in the center with a fork to prevent puffing.

Chill the cookies in the refrigerator for at least 20 minutes (or up to 2 days, covered with plastic wrap).

Preheat the oven to 300°F. Line a half-sheet pan with parchment paper.

Place the chilled cookies 2 inches apart on the prepared pan (about 12 per batch). Bake for 20 to 23 minutes, until the bottoms are just lightly golden and the tops remain pale.

Let the cookies cool on the pan for 5 minutes, then transfer to a wire rack to cool completely.

In a mini food processor, combine the powdered sugar and freeze-dried strawberries. Process until the strawberries are powdery and the mixture is pale pink.

Transfer to a small bowl. Add 3 tablespoons of milk and stir until smooth and creamy. For a thinner glaze, add more milk 1/2 teaspoon at a time.

Dip the top of each cooled cookie into the glaze, pressing gently to coat. Lift and shake off excess. Place glazed cookies back on the wire rack to set.