1. Heat oven to 185 degrees. Cut a 7-inch circle from a 10-inch square of 1/4-inch-thick cardboard or Foamcore. Set circle aside to use for bottom of cake, and reserve the cut-out square as a template. Cut three 11-by-17-inch rectangles out of parchment paper, and set parchment aside.

2. Fill a medium saucepan with 2 inches of water, and bring to a simmer. Place sugar and egg whites in the bowl of an electric mixer. Hold bowl over simmering water, and whisk until warm to the touch and sugar has dissolved, about 3 minutes.

3. Using the whisk attachment, beat egg whites on medium speed until stiff and glossy, about 4 minutes; do not overbeat.

4. Place the template over one end of a piece of parchment. Scoop 1 cup egg-white mixture into the center. Using an offset spatula, spread the mixture into an even layer. Remove the template. Make another disk on other end of parchment. Create three more disks on remaining two pieces of parchment. Slide each piece of parchment onto a 11-by-17-inch baking sheet.

5. Transfer the meringue disks to the oven, and bake 1 1/2 hours, making sure they don't brown. Turn off heat, and let the disks dry completely in oven, 6 hours to overnight.

6. Place one meringue disk on the reserved 7-inch cake round. Using a gelato paddle or a large spoon, build up an even layer of mango sorbet on top of the disk. Top with a second meringue disk. Continue the layering process, returning cake to the freezer to harden, as needed. Use remaining grapefruit, raspberry, and green-apple sorbets and the meringue disks. Top with a disk. Transfer cake to freezer, and freeze until hardened. Let soften slightly before serving. Serve with a spoon in bowls.