Frosting
Chill bowl of stand mixer and whisk attachment or large metal bowl and beaters for at least 15 minutes.

In small saucepan over moderate heat, bring 1 cup cream to simmer. Transfer white chocolate to medium heatproof bowl, pour hot cream over, and whisk until smooth. Whisk in rum and salt. Let cool at room temperature until thickened slightly, about 1 hour.

In chilled bowl of electric mixer fitted with whisk attachment, beat remaining 2 cups cream at moderately high speed until whisk leaves marks but cream does not quite hold soft peaks, 6 to 8 minutes. Turn mixer off, then add white chocolate mixture and beat just until stiff peaks begin to form, about 5 minutes. (Do not overbeat, or cream will curdle.) Refrigerate until firm, about 3 hours. (Frosting can be made ahead and refrigerated, covered, up to 8 hours.)

While frosting is chilling, make cake
Preheat oven to 350° F. Line bottoms of cake pans with parchment paper.

In large bowl, whisk together almond and coconut flours.

In bowl of electric mixer fitted with whisk attachment, beat egg yolks at high speed until pale yellow and fluffy, 2 to 3 minutes. Reduce speed to moderately low and beat in rum and all but 1 tablespoon confectioner's sugar. Scrape down bowl, then increase speed to high and beat until pale and thick, about 1 minute. Reduce speed to low and gradually add almond and coconut flour mixture, scraping down bowl and folding in last of flour by hand. Set aside.

In clean dry bowl of electric mixer fitted with whisk attachment, beat egg whites on moderate speed until very foamy, about 1 minute. Beat in cream of tartar, salt and remaining 1 tablespoon confectioner's sugar. Increase speed to moderately high and beat until whites hold stiff peaks, about 2 minutes. Fold 1 cup beaten egg whites into yolk mixture to lighten, then gently fold in remaining whites. Divide batter between pans, smoothing tops, and bake until layers are golden brown and toothpick inserted in center comes out clean, 20 to 25 minutes. Invert pans onto wire racks and cool completely, at least 1 hour. Run knife around cake layers to loosen, invert onto rack, and peel off the parchment.

Assemble cake
Layer cake, using 1 cup frosting between each layer, then frost top and sides with remaining frosting. Press coconut flakes onto the frosting. Refrigerate until frosting is firm, at least 3 hours or overnight.

Tomato sauce
In a 4-quart nonreactive saucepan over very low heat, stir together tomatoes and oil. Bring to simmer, cover partially, and continue simmering, stirring occasionally, until sauce is reduced to 1 cup, 20 to 25 minutes. Stir in sugar, salt and oregano, cover. Keep warm or refrigerate, covered, up to 5 days.

Pizza crust
In bowl of electric mixer, whisk together tapioca flour, white rice flour, chickpea flour, sorghum flour, xanthum gum and salt.

In small saucepan over moderate heat, stir together milk and 1/4 cup water and heat until warm but not hot to the touch, about 1 minute (the mixture should register between 105° F and 115° F on candy thermometer). Stir in yeast and sugar. Add milk–yeast mixture, egg whites and 2 tablespoons oil to dry ingredients and, using paddle attachment, beat at medium speed, scraping bowl occasionally, until dough is very smooth and very thick, about 5 minutes.

Remove racks from oven, set pizza stone or heavy upturned baking sheet on bottom of oven, and preheat to 400° F. (Preheat at least 45 minutes if using pizza stone or 20 minutes if using baking sheet.)

Have ready two 12-inch squares parchment paper. Scrape half of dough onto each square and form each half into a ball. Coat each ball with 2 teaspoons oil, then use oiled fingertips to pat and stretch each ball into 9-inch-diameter round, 1/4 inch thick, with a 1/2-inch-thick border. Loosely cover rounds with plastic wrap and let rise in warm draft-free place until each pizza is about 10 inches in diameter, about 20 minutes.

Using baking peel, transfer 1 crust with parchment to preheated pizza stone and bake until top is puffed and firm and underside is crisp, 5 to 10 minutes. Using baking peel and discarding parchment paper, transfer baked crust to rack to cool. Bake second crust in same manner. (Baked crusts can be made ahead and frozen, wrapped in plastic wrap, up to 1 month. Thaw in 350° F oven until hot, 4 to 5 minutes, before topping and broiling.)

Top and broil pizzas
Preheat broiler. Transfer baked crusts to 2 large baking sheets. Brush 1 teaspoon olive oil over each crust. Spread each with sauce, leaving 1/2-inch border bare, then sprinkle each with mozzarella and Parmigiano-Reggiano. Drizzle remaining 2 teaspoons olive oil over pizzas.

Broil pizzas about 4 inches from heat, rotating as needed for even browning, until cheese is bubbling and browned in places and crust is golden brown, 4 to 8 minutes. Scatter with basil, slice, and serve immediately.