1. Coat two 8-inch springform pans with cooking spray. Line sides with parchment, leaving a 2-inch collar extending over top.

2. Puree blackberries, blueberries, and 2 tablespoons lemon juice in a blender until smooth. Strain through a fine sieve; discard solids.

3. Whisk 2 egg whites and a pinch of salt with a mixer on medium speed until foamy. Meanwhile, combine 1/2 cup plus 1 tablespoon sugar with 3 tablespoons water in a small saucepan over medium-high heat, stirring until sugar dissolves. Cook until syrup reaches 250 degrees on a candy thermometer. Immediately add sugar syrup in a slow, steady stream down the side of the bowl. Raise speed to high, and whisk until thick, shiny, and cooled, about 7 minutes. Whisk in berry puree. Pour mixture into 1 pan, and freeze until firm, about 2 hours.

4. Meanwhile, puree papayas and 2 tablespoons lemon juice in a blender until smooth. Strain.

5. Repeat step 3 with 2 egg whites, a pinch of salt, 1/2 cup plus 1 tablespoon sugar, and 3 tablespoons water, whisking in papaya puree. Pour mixture into remaining pan, and freeze until firm, about 2 hours.

6. Puree kiwifruits and 2 tablespoons lemon juice in a blender until smooth. Strain.

7. Repeat step 3 with 2 egg whites, a pinch of salt, 1/2 cup plus 1 tablespoon sugar, and 3 tablespoons water, whisking in kiwifruit puree. Pour mixture onto frozen berry layer, and freeze until firm, about 2 hours.

8. Meanwhile, puree strawberries, raspberries, and 2 tablespoons lemon juice in a blender until smooth. Strain.

9. Repeat step 3 with 2 egg whites, a pinch of salt, 1/2 cup plus 1 tablespoon sugar, and 3 tablespoons water, whisking in strawberry-raspberry puree. Pour mixture onto frozen papaya layer, and freeze until firm, about 2 hours.

10. Puree apricots and 2 tablespoons lemon juice in a blender until smooth. Strain.

11. Repeat step 3 with 2 egg whites, a pinch of salt, 1/2 cup plus 1 tablespoon sugar, and 3 tablespoons water, whisking in apricot puree. Pour mixture onto frozen kiwi layer, and freeze until firm, about 2 hours.

12. Release sides of pans; carefully transfer papaya and strawberry-raspberry layer cake onto apricot layer. Freeze cake while you make the frosting.

13. Remove parchment paper. Frost top and sides of cake. Freeze until ready to serve.