Heat the oven to 350°F (175°C). Line two baking sheets with parchment. Whisk flour, cocoa, baking soda, and salt in a bowl.
Beat butter and both sugars until creamy. Beat in egg and vanilla, then mix in dry ingredients just until a soft dough forms.
Divide into 24 equal portions, about 1 tablespoon each. Roll into balls, arrange 2 inches apart, and flatten to similar thickness.
Bake one sheet at a time for 8–10 minutes, until edges set and centers remain soft. Rest on the sheet 5 minutes, then cool fully on a rack.
Pair cookies by size and chill on a tray for 30 minutes. Let ice cream soften only enough to spread rather than melt into liquid.
Place about 1/4 cup ice cream on the flat side of one cookie. Top with its partner and press gently to distribute filling to the edges.
Wrap each sandwich individually and freeze immediately. Work in small groups so waiting ice cream and assembled sandwiches do not become warm.
Freeze at least 2 hours before serving. Let a sandwich stand 2–3 minutes if very hard, keeping the remaining portions frozen until needed.
