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.
Notes
It's easy to space out the process of these cookies over several days if you want to. The dough can stay in the fridge for up to 2 days before rolling, cutting, and baking the cookies. The baked cookies also keep well, covered, overnight, if you want to decorate them the next day.
If you have leftover frosting, it can be covered and stored in the fridge for up to 5 days.