Preheat oven to 350°F (180°C). In a small bowl, whisk flour with baking soda and salt. Using an electric mixer, in a large bowl, beat peanut butter with butter until smooth. Beat in brown sugar then egg and vanilla, scraping down sides to mix well. Gradually beat in flour mixture.
If needed, flour your hands then roll dough into 1 1/2–inch (4 cm) balls. Place at least 2 inches (5 cm) apart on ungreased baking sheets. Flour the base of a large glass and press on each ball to flatten cookie to about 3 inches (7.5 cm). Using floured tines of a fork, make criss–crosses gently on top of cookies.
Bake in centre of preheated oven until golden and barely set and edges are brown, 10 to 12 minutes. Remove from oven and leave on baking sheet 1 minute, then transfer to a rack to cool completely.
For easier spreading, soften ice cream slightly. Top flat side of one cookie with about 1/2 cup (125 mL) ice cream. Then place flat side of another cookie on top, sandwiching ice cream. Unfilled cookies will last well in an airtight container for about 5 days, or freeze. Ice cream sandwiches should be wrapped in plastic wrap then overwrapped with foil.