Special equipment: a 4-quart heat-safe bowl, 2.5-quart heat-safe bowl
Position a rack in the bottom half of the oven and preheat to 350 degrees F. Cut two 16-by-6-inch pieces of parchment. Butter and flour the inside of a 4-quart heat-safe bowl, press a piece of parchment paper against the bottom and up the side of the bowl. Press the remaining parchment on top of the first, turning it slightly so that the 2 pieces form an "X"; set aside. Butter and flour the bottom of a 2.5-quart heat-safe bowl; set aside.
Preheat the oven to 350 degrees F. Whisk together the flour, baking soda and salt in a large bowl. Beat together the butter, brown sugar and granulated sugar in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 4 minutes. Add the eggs, one at a time, beating to incorporate after each addition. Beat in the vanilla. Reduce the mixer speed to low and add the flour mixture, beating just until incorporated. Add the chocolate chips and mix until combined.
Put the dough into the larger prepared bowl. With generously floured hands, press the dough evenly up the side and across the bottom of the bowl (re-flour your hands as needed). Place the smaller bowl inside the larger bowl so that the buttered and floured bottom-side sits directly on top of the cookie dough. Bake until the cookie is a deep golden brown, 50 to 55 minutes. Let the cookie cool completely before removing the inner bowl, about 2 hours.
To fill: Brush ice cream topping evenly over the inside of the cookie. Stack the peanut butter-filled chocolate cups in a tower in the middle of the cookie bowl. Let harden completely in the freezer, about 15 minutes.
Spread half the ice cream in the bowl, working around the chocolate cup tower. Layer with 1 1/2 cups chopped candy bars then spread the remaining ice cream on top. Sprinkle with the remaining candy bars and cover with plastic wrap. Freeze until solid, 6 hours or up to overnight.
Remove from the freezer 15 minutes before serving. Flip the cookie bombe upside down on a serving platter and lift off the bowl and parchment strips. Decorate with whipped cream and sprinkles around the base and top of the bombe. Slice with a very hot, sharp knife as you would a cheesecake.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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