Preheat oven to 350 degrees, with rack in center. In a medium bowl, whisk together flour, baking soda, and salt. Using an electric mixer, beat butter and sugars until light and fluffy. Add sour cream, egg, and vanilla; beat until smooth. With mixer on low, gradually add flour mixture; mix in 1 cup chocolate chips.
Using 1 rounded tablespoon per cookie, drop mounds of dough, about 1 1/2 inches apart, onto two baking sheets. (You'll have 25 to 30 cookies.) Bake until golden, 10 to 12 minutes; transfer to a rack to cool completely.
Scatter remaining chocolate chips on a plate. Place about 2 tablespoons ice cream between 2 cookies, bottoms facing in; press gently. Roll the sides of each sandwich in chips. (If ice cream is melting, place sandwiches in freezer until firm enough to roll in chips.) Transfer sandwiches onto baking sheets. Freeze until firm, at least 2 hours; then wrap individually in plastic.
Freeze up to 2 weeks.