Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together flour, cocoa powder, baking soda, and salt.
In a large bowl, beat butter and sugar until creamy. Add egg and vanilla; beat until combined.
Gradually mix in dry ingredients until a dough forms.
Scoop dough into small balls and flatten slightly. Place on baking sheets.
Bake for 8-10 minutes. Let cool completely.
Scoop softened ice cream onto one cookie, top with another to form a sandwich.
Wrap each sandwich in plastic wrap and freeze until firm, about 2 hours.