Place cookies in a food processor; process until finely ground. Add 1/3 cup brown sugar and butter; pulse 2 to 3 times or just until combined. Press crumb mixture into bottom and up sides of a 9-inch pie plate coated with cooking spray. Bake at 350° for 10 minutes; cool completely on a wire rack.
Place 3/4 cup brown sugar, cream cheese, peanut butter, and vanilla in a mixing bowl; beat with a mixer at medium speed until smooth. Fold in whipped topping. Arrange banana in bottom of prepared crust. Spread peanut butter mixture over banana; drizzle with syrup. Cover and freeze 8 hours; let stand at room temperature 15 minutes before serving.