To make the cookie dough, combine flour, cocoa powder, and salt in a medium bowl. Whisk to blend. Set aside.
Combine butter and powdered sugar in a large bowl, and beat with an electric mixer on medium-high speed for 2-3 minutes, until light and fluffy. Blend in the egg and vanilla.
Add dry ingredients and beat on low speed, just until incorporated and no streaks remain. Form the dough into a disc, wrap tightly with plastic wrap, and refrigerate until firm, at least 1 hour.
When ready to bake, preheat the oven to 325 degrees F (170 C) and line baking sheets with silicone mats or parchment paper. On a lightly-floured work surface, roll out the dough to about ¼-inch thickness. Cut out desired shapes with cookie cutters. Place cut-outs on the prepared baking sheets. Bake 10-12 minutes, just until set. Let cookies cool on the sheet for 5-10 minutes, then transfer to a wire rack to cool completely.
To make the frosting, place the freeze-dried strawberries in a powerful blender or food processor, and process into a powdery crumb. You should have around 1/4 cup. Set aside.
In a large bowl using an electric mixer, beat butter on medium-high speed for about 2 minutes. Add powdered sugar, strawberry powder, cream, and vanilla. Beat on low speed for 30 seconds, then on high for 2 minutes, until frosting is thick and spreadable. Taste and add a pinch of salt if the frosting is overly sweet.
Spread 2-3 tablespoons frosting on each cookie. After about 20 minutes, the frosting will form a light crust, and the cookies can be stacked and stored either at room temperature or in the fridge. They'll keep well, tightly wrapped, for 2-3 days.