Preheat your oven to 250°F (120°C) and line two baking sheets with parchment paper. In a clean, dry mixing bowl, whip the large egg whites on medium speed until foamy. Gradually add cream of tartar, then slowly incorporate granulated sugar until stiff peaks form. Gently fold in ground pistachios, powdered sugar, and cake flour until combined, ensuring a light and airy mixture.

Transfer the meringue mixture to a piping bag fitted with a large round tip. Pipe the meringue into 7-inch circles onto the prepared parchment, creating three evenly spaced layers. Bake in your preheated oven for about 1 hour, or until the meringues are crisp and can be easily lifted from the parchment. Turn off the oven, crack the door open slightly, and allow the meringues to cool completely inside.

While the meringue layers cool, prepare the pistachio buttercream. In a saucepan, combine water and granulated sugar, and bring to a simmer to create a sugar syrup. Meanwhile, in a mixing bowl, beat the large egg yolks until thick and pale. With the mixer running, slowly drizzle in the hot syrup, whisking continuously until the mixture has cooled slightly and thickened. Gradually beat in room temperature unsalted butter, followed by vanilla extract, pistachio paste, and salt until smooth and creamy.

Begin assembling your Pistachio Dacquoise Cake by placing one meringue layer on a serving platter. Spread a generous layer of pistachio buttercream on top, then repeat the process with the remaining meringue layers and buttercream, finishing with a layer of buttercream on top. Be sure to level the layers well for an elegant finish.

Once the cake is assembled, refrigerate it for a couple of hours to allow the layers to set and flavors to meld beautifully. Before serving, let the cake sit at room temperature for about 20 minutes for the best texture. Slice into this luscious Pistachio Dacquoise Cake and enjoy.