1. Preheat oven to 375ºF. Line 2 large baking sheets with parchment.
2. Sift flour, baking soda and salt together. Using an electric mixer on medium speed, beat butter and sugar together until light, about 2 minutes. Add egg; mix well. Beat in peanut butter and vanilla. Beat flour mixture into butter mixture.
3. Roll dough into 1 1/2-inch balls; place 2 inches apart on sheets. Flatten with a fork in a crisscross design. Bake until golden, 12 to 15 minutes. Let cool on sheets for 5 minutes; transfer to a rack to cool.
4. Place 1 scoop of ice cream on a cookie, top with another cookie and carefully press down. Wrap in plastic wrap and freeze. Repeat with remaining cookies and ice cream. Freeze for 45 minutes; serve.