Remove cake from pan and cut horizontally into 4 slices. Line pan with plastic wrap, leaving 6-inch overhang on two sides. Build cake layers in pan in this order: bottom cake slice, lime sherbet, cake slice, Oreo crumbs, vanilla ice cream, cake slice, raspberry sherbet, top cake slice. Wrap in plastic and freeze for a minimum of 1 hour (or up to 3 days).
In a heatproof bowl, lightly whisk together egg whites and pinch of cream of tartar. Set bowl over a pan of simmering water and whisk until whites are foamy. Slowly whisk in sugar and cook, whisking, until sugar is dissolved.
Remove bowl from heat and, with an electric mixer, beat on medium-high speed until stiff, glossy peaks form, about 6 minutes. Add vanilla and beat for 1 minute. Remove cake from pan and dollup meringue over top.