Process blackberries and raspberries in a food processor until very smooth. Pour through a fine wire-mesh strainer, using the back of a spoon to press as much pulp through as possible; discard solids. Cover and chill 2 hours to 2 days.
Preheat oven to 350°F. Pulse waffle cone pieces, peanuts, sugar, and salt in a food processor until coarsely ground. Add butter, and pulse until well combined. Using a straight-sided glass, press the mixture into the bottom and all the way up the sides of a 9-inch springform pan.
Bake in preheated oven until fragrant, 10 to 12 minutes. Cool completely, about 30 minutes. Freeze 30 minutes.
Beat malt powder and 1 quart of gelato with an electric mixer at low speed until smooth and soft, about 1 minute. In a separate bowl, beat chilled berry puree and remaining quart of gelato at low speed until blended, about 1 minute.
Spoon some of the berry-gelato mixture into prepared pan. Top that with some of the malted gelato. Repeat layers, ending with the berry-gelato mixture. Using a wooden skewer, swirl layers together. Cover and freeze 24 hours. Remove sides of pan. Just before serving, top cake with berries and, if desired, a sprinkling of powdered sugar.