
Coat two 8-inch spring form pans with coconut oil
Line the sides with parchment, leaving a 2-inch collar extending over top.
Puree Blackberries, blueberries, and 2 tbsp. lemon juice in a blender until smooth
Strain through a fine sieve; discard solids.
Whisk 2 egg whites and a pinch of salt with a mixer on medium speed until foamy.
Meanwhile, combine ½ cup plus 1 tbsp. sugar with 3 tbsp. water in a small saucepan over medium-high heat, stirring until sugar dissolves.
Cook until syrup reaches 250o. 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.
Meanwhile, puree papayas and 2 tbsp. lemon juice in a blender until smooth. Strain.
Repeat step 3, with 2 egg whites, a pinch of salt, ½ cup plus 1 tbsp. sugar, and 3 tbsp. water, whisking in papaya puree. Pour mixture into remaining pan, and freeze until firm, about 2 hours.
Puree kiwifruits and 2 tbsp. lemon juice in a blender until smooth. Strain.
Repeat step 3 with 2 egg whites, a pinch of salt, ½ cup plus 1 tbsp. sugar, and 3 tbsp. water, whisking in kiwifruit puree. Pour mixture onto frozen berry layer, and freeze until firm, about 2 hours.
Meanwhile, puree strawberries, raspberries, and 2 tbsp. lemon juice in a blender until smooth. Strain.
Repeat step 3 with 2 eggs whites, a pinch of salt, ½ cup plus 1 tbsp. sugar, and 3 tbsp. water, whisking in strawberry-raspberry puree. Pour mixture onto frozen papaya layer, and freeze until firm, about 2 hours.
Puree apricots and 2 tbsp. lemon juice in a blender until smooth. Strain.
Repeat step 3 with 2 egg whites, a pinch of salt, ½ cup plus 1 tbsp. sugar, and 3 tbsp. water, whisking in apricot puree. Pour mixture onto frozen kiwi layer, and freeze until firm, about 2 hours.
Release sides of pans; carefully transfer papaya and strawberry-raspberry layer cake onto apricot layer. Freeze cake while you make the frosting.
Remove parchment paper. Frost top and sides of cake. Freeze until ready to serve.
Bring ½ cup sugar & the water to a boil in a saucepan, stirring until sugar dissolves. Continue to cook, without stirring, washing down sides of pan with a wet pastry brush to prevent crystals from forming, until mixture reaches 230o.
Meanwhile, whisk egg whites with a whisker on medium speed until soft peaks form. Gradually whisk in remaining 2 tbsp. sugar. Immediately add sugar syrup in a slow, steady stream down the side of the bowl once it reaches 230o. Raise speed to high, and whisk until mixture is thick, shiny, and cooled, about 7 minutes. Use immediately.