1. Preheat oven to 350ºF. Line 2 rimmed baking sheets with parchment. Beat butter and both sugars until light and fluffy, about 3 minutes. Beat in egg and vanilla. In a bowl, whisk together flour, baking soda and salt. Gradually stir dry ingredients into butter mixture until combined. Stir in oats.
2. Drop dough by heaping tablespoonfuls onto baking sheets. Bake until cookies are light brown, about 14 minutes. Let cool for 3 minutes, then transfer cookies to wire racks to cool completely. Repeat with remaining dough.
3. Arrange 8 cookies, bottom side up, and top each with a small scoop of ice cream. Top with a second cookie, bottom side down, pressing together slightly. Wrap each in plastic and freeze until firm, about 1 hour.