Preheat the oven to 300 degrees F (150 degrees C). Line four 8x12-inch pans with parchment paper.

Beat egg whites in a glass or metal bowl until foamy; sprinkle with cream of tartar. Gradually add 2 cups white sugar, continuing to beat until stiff peaks form. Fold in 2 cups chopped cashews. Divide meringue between four pans; spread evenly to edges.

Bake in the preheated oven until light golden brown, 30 minutes. Remove meringues from the baking pans while still hot; let cool. When cool, trim edges so all 4 meringue layers uniformly shaped. Set aside.

Beat butter in a large bowl until light and fluffy. Whip egg yolks in a separate bowl until thick and lemon colored.

Meanwhile, combine remaining 2 cups sugar and 1 cup water in a saucepan; bring to a boil. Heat to 270 to 290 degrees F (132 to 143 degrees C) or until a small amount of syrup dropped into cold water forms hard but pliable threads. Pour hot syrup in a thin stream into beaten egg yolks, while beating at high speed. Beat in whipped butter until buttercream smooth. Beat in rum.

Arrange 1 layer meringue on a serving plate; spread approximately 1/4 buttercream evenly to edges. Top with 1 more meringue layer; repeat spreading and topping with remaining buttercream and meringue layers, ending with buttercream. Smooth sides with spatula. Sprinkle top with remaining 2 cups chopped cashews. Freeze until solid. Cut 1 inch slices to serve frozen dessert.