Preheat oven to 350 degrees F. Line a large baking sheet with parchment paper.
In a large bowl with an electric mixer, mix butter, peanut butter and both sugars on medium speed, until well blended. Add egg and beat well. In a separate bowl, whisk flour, baking soda, baking powder and salt. Add to butter mixture, blending well.
Scoop walnut-sized (about 1-inch) balls of dough and place on prepared baking sheet. Gently flatten with a fork dipped in sugar, making a crisscross pattern on each.
Bake 8 minutes. Let rest on pan for a few minutes, then remove to a wire rack. Cool completely before adding ice cream.
For the banana ice cream:
Combine ice cream and banana until well blended. Scoop 1 tablespoon ice cream onto 1 cookie. Top with another cookie, gently pressing together to make a sandwich. Place on a baking sheet and freeze for 1 hour, or until firm.
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