Preheat your oven to 200°F (93°C). In a clean mixing bowl, whisk 5 large egg whites until frothy. Gradually add ⅝ cup granulated sugar, 1 teaspoon cream of tartar, and ¼ teaspoon salt. Continue whisking until stiff peaks form and the mixture is glossy.

Line two 7-inch round cake pans with parchment paper. Using a piping bag, pipe the meringue mixture into two even circles on the prepared pans. Smooth the tops with a spatula. Bake in the preheated oven for 1 to 2 hours, or until the meringue is dry to the touch.

In a saucepan, combine ¾ cup granulated sugar and ⅓ cup water over medium heat. Stir until sugar dissolves, then bring the mixture to a boil. Meanwhile, whisk 8 egg yolks in a mixing bowl until pale and fluffy. Once the sugar syrup reaches 240°F (115°C), carefully pour it into the yolks while whisking continuously until the mixture is thick.

Once cooled, gradually add 2½ sticks of softened unsalted butter, mixing until creamy and smooth. Add 1 teaspoon vanilla extract, 4 tablespoons pistachio paste, and ¼ teaspoon salt. Beat until fluffy and well combined.

On a serving platter, place one meringue layer and spread a generous layer of pistachio buttercream over the top. Add the second layer and repeat the process, finishing with a smooth layer of buttercream on top.

Refrigerate for at least 20 minutes to allow flavors to meld and the buttercream to set. Slice into elegant pieces and serve.