Make cookies: Preheat oven to 350°F. Line 2 baking sheets with parchment paper. Sift flour, cocoa, baking soda and salt into a bowl.
Combine butter and sugar in a large mixing bowl; cream together with an electric mixer until fluffy. Add egg and vanilla, and beat until smooth. Add flour mixture and mix until just combined. Stir in chocolate chips.
Scoop up heaping tablespoonfuls of dough and roll into 1 1/2- to 2-inch balls. Place balls on baking sheets 3 inches apart (you'll have about 8 balls per sheet). Bake until dry on top, 13 to 15 minutes. Let cool on baking sheets on wire racks.
Make sandwiches: Soften ice cream at room temperature for 3 minutes. Set containers on their sides on a cutting board and use a sharp serrated knife to cut each pint of ice cream into 7 circles, cutting through cardboard. Peel cardboard away and place each disk of ice cream between 2 cookies. Place sandwiches on a baking sheet, cover with plastic and freeze until firm, at least 1 hour and up to 2 days.