Position racks in top and bottom third of oven; preheat to 350°F. Line 2 large rimmed baking sheets with parchment. Using electric mixer, beat first 5 ingredients until fluffy. Beat in flour.
Place 1 cup powdered sugar in bowl. Working with 1 level tablespoon at a time, shape dough into balls; roll in powdered sugar to coat. Place on sheet. Press each lightly with flat-bottomed cup. Sprinkle each with more powdered sugar, then press again to thin 3 1/4-inch round. Bake cookies 8 minutes. Reverse sheets; bake until golden at edges, about 9 minutes longer.
Meanwhile, combine powdered sugar and freeze dried strawberries in the bowl of a food processor. Process for a minute or two or until strawberries are powderized. Pour mixture into a medium size bowl. Add milk and stir until smooth, adding more milk by 1/2 teaspoonfuls if glaze is too thick to spread. If too thin, add a bit more powdered sugar.
Place about teaspoon of icing in the center of a cookie. Spread glaze over cookies to the edges with the back of a spoon. Place on cooling rack to dry for at least one hour.
*~ If you can't find freeze-dried strawberries just finely mash fresh strawberries to equal 1/4 cup. Add mashed strawberries instead of the milk and adjust with a bit of milk or extra powdered sugar to achieve a thick but pourable consistency.