Fruit purees can be refrigerated for up to 2 days; frosted cake can be frozen overnight. Food-safety note: The egg whites in this cake are not fully cooked.
STEP 1
Coat two 8-inch springform pans with cooking spray. Line sides with parchment, leaving a 2-inch collar extending over top.
STEP 2
Puree blackberries, blueberries, and 2 tablespoons lemon juice in a blender until smooth. Strain through a fine sieve; discard solids.
STEP 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.
STEP 4
Meanwhile, puree papayas and 2 tablespoons lemon juice in a blender until smooth. Strain.
STEP 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 papayapuree. Pour mixture into remaining pan, and freeze until firm, about 2 hours.
STEP 6
Puree kiwifruits and 2 tablespoons lemon juice in a blender until smooth. Strain.
STEP 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 froze
n berry layer, and freeze until firm, about 2 hours.
STEP 8
Meanwhile, puree strawberries, raspberries, and 2 tablespoons lemon juice in a blender until smooth. Strain.
STEP 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 papayalayer, and freeze until firm, about 2 hours.
STEP 10
Puree apricots and 2 tablespoons lemon juice in a blender until smooth. Strain.
STEP 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 apricotpuree. Pour mixture onto frozen kiwi layer, and freeze until firm, about 2 hours.
STEP 12
Release sides of pans; carefully transfer papaya and strawberry-raspberry layer cake onto apricot layer. Freeze cake while you make the frosting.
STEP 13
Remove parchment paper. Frost top and sides of cake. Freeze until ready to serve.