Place racks in upper and lower thirds of oven; preheat to 350°. Toast 1 cup (140 grams) pistachios on a rimmed baking sheet, tossing halfway through, until golden brown, 10–13 minutes. Let cool, then coarsely chop; set aside for decorating.
Reduce oven temperature to 300°. Cut 2 large sheets of parchment paper and trace three 7x3½" rectangles with a pencil on each sheet. Turn parchment over and place each on an 18x13" baking sheet (you should be able to see the traced rectangles through the parchment; they will be your guides later). Pulse powdered sugar and remaining 1 cup (140 g) pistachios in a food processor until the consistency of very coarse sand. Transfer to a large bowl.
Place raw egg whites in the bowl of a stand mixer fitted with the whisk attachment. Beat on medium speed until frothy, about 2 minutes. With the motor running, gradually add granulated sugar, meringue powder (if using), and salt. Increase speed to medium-high; beat until stiff peaks form and meringue is glossy, about 7 minutes. Fold into pistachio mixture in thirds, folding until just combined before adding more. (It should still look fluffy.)
Transfer batter to a piping bag fitted with a ½"-wide tip (or fill a resealable plastic bag and snip off a corner) and pipe to fill each traced rectangle on prepared baking sheets. (Or, you can divide batter into 6 portions and spread to fill each rectangle all the way to the edges with an offset spatula.)
Bake, rotating halfway through, until meringues are light golden brown, dry to the touch, and firm (they shouldn’t crack as soon as you touch them), 40–50 minutes. Turn off oven, prop door open (a wooden spoon works well), and let sit in oven 20 minutes.
Remove meringues from oven and carefully peel away parchment while meringues are still warm. Let meringues cool completely.
While the meringues cool, peel 2 mangoes and cut flesh from pits. Purée in a blender or food processor until smooth. Transfer to a small saucepan; add lemon juice. Cook over low heat until slightly thickened, 10–15 minutes. (You should have about 1½ cups.) Let cool slightly.
With motor on medium-high speed, beat egg yolks in the clean bowl of a stand mixer fitted with whisk attachment until pale and doubled in volume, about 8 minutes.
Meanwhile, place granulated sugar and 6 Tbsp. water in another small saucepan (do not stir) and clip thermometer to side of pan. Cook over low heat until thermometer registers 240°, about 8 minutes.
With motor on high speed, carefully stream hot sugar into whipped egg yolks, avoiding whisk and sides of bowl, and beat until bottom of bowl is slightly cooler to the touch, about 4 minutes. Mixture should be very pale, smooth, and glossy with soft peaks that fall back onto themselves.
Turn off mixer and add warm mango purée. Beat on medium-high just to combine. Reduce speed to medium and, with motor running, add butter 1 Tbsp. at a time, beating until incorporated and smooth after each addition. Add a little more lemon juice to make a more tart buttercream if desired.
Take 1 meringue and dot a little buttercream in several spots on the bottom. Set on a parchment-lined baking sheet and press gently to adhere. Using an offset spatula, spread ¾ cup buttercream over. Place another meringue on top and spread another layer of buttercream over. Repeat process until all meringues are stacked, pressing down a bit firmly when placing each meringue to maintain even layers.
Using spatula, spread buttercream around sides and top of stacked meringues. Press reserved pistachios onto sides as desired, saving some for decorating the top. (If your buttercream is very soft, chill cake 10 minutes before pressing on nuts.)
Slice remaining mango and arrange decoratively on top of cake. Fill in gaps with remaining pistachios. Freeze at least 2 hours before serving.
Do ahead: Cake can be made 1 week ahead. Wrap tightly once firm; keep frozen.