Using a blender, pulverize freeze-dried strawberries into a fine powder.
Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
Cream butter, granulated sugar, and vanilla extract together until light and fluffy. Beat in the egg until well combined. Gradually incorporate flour, baking soda, part of the strawberry powder, and salt, mixing until just combined. Fold in the strawberry jam.
Roll the dough into balls and place them on the prepared baking sheet.
Bake for about 8 minutes. Once lightly golden on the edges, tap the baking sheet gently to flatten them slightly and then bake for an additional 2–3 minutes.
In a bowl, combine cream cheese, powdered sugar, hot milk, vanilla extract, and the remaining strawberry powder. Mix until smooth.
Dip cooled cookies into prepared icing or drizzle icing over the tops as you prefer.