For the cookie: Position an oven rack in the lowest position in the oven and preheat to 375 degrees F. Thoroughly grease a 10-inch cast-iron skillet with butter (or use a stainless-steel skillet coated with nonstick vegetable oil spray).
Mix the brown sugar, granulated sugar, butter and salt in the bowl of a stand mixer fitted with the paddle attachment until no large lumps remain, about 1 minute.
Stir together the egg and vanilla in a large bowl. Fold in the banana and white chocolate. Combine the flour and baking soda in a separate large bowl until thoroughly combined. Add the egg mixture and butter mixture to the flour mixture and thoroughly combine. Fold in the pretzels until evenly distributed.
Scrape the batter into the prepared skillet and pat into an even layer with damp hands so the batter extends to the edges of the skillet.
Bake until the outer edges are browned and slightly puffed, 20 to 25 minutes (15 to 18 minutes for stainless steel). Let cool.
For the whipped cream: Beat the heavy cream and peanut butter in a large cold mixing bowl with an electric mixer until it starts to thicken, about 1 minute. Gradually beat in the powdered sugar, then continue beating until stiff peaks form (when you pull the beater up, peaks will form and not fall over), 3 to 4 minutes.
Slice and serve the cookie with the whipped cream.