Line a large baking sheet with parchment paper and place two 7-inch pastry rings on it, ungrased. Preheat the oven to 465°F / 240°C.
Add the pistachios to a food processor and grind them up until as finely as you can before they start to release their oils. Don't go too far or you'll end up with pistachio paste!
Add the flour and powdered sugar to the ground pistachios and pulse until everything is combined. Set aside.
Make sure the egg whites are room temperature, then whip them up with the cream of tartar and a pinch of salt in a stand mixer or with an electric whisk.
Once the egg whites start to get frothy, start adding the sugar, one tablespoon at a time, whipping for 30-60 seconds in between each addition.
Once all the sugar is in, the meringue should be thick and glossy with no more grains of sugar in it. Keep whipping until it reaches very stiff peaks.
Once the meringue is stiff and glossy, add half of the pistachio flour you made earlier and gently fold it into the meringue. Be careful not to knock too much air out of it.
Once incorporated, add the reamining pistachio flour and fold that in completely too.
Divide the dacquoise batter evenly between the two pastry rings and spread it out as flat as possible.
Put the dacquoise into the oven and immediately turn the temperature down to 350°F / 180°C. Bake for 15 minutes.
After 15 minutes, turn the oven off and open the oven door slightly. Let the dacquoise sit in there for an hour before moving to the countertop and letting them cool completely in the pastry rings.
Once completely cooled, run a sharp knife around the edge of the pastry rings and gently release the dacquoise.
Make the French Buttercream
Place the egg yolks into the bowl of a stand mixer with a whisk attachment fitted. Whip them on medium-high speed until they are very pale in color (about the color of butter). This should take about 5 minutes.
Meanwhile, heat the sugar and water in a small saucepan on medium heat and bring to a boil. Use a candy thermometer to track the temperature, we want it to get to exactly 240°F / 115°C.
Once the sugar syrup reaches the right temperature, turn the mixer down to low speed and carefully pour the sugar syrup into the egg yolks in a slow, steady stream. Be careful not to hit the sides of the bowl or the whisk as you do it.
The bowl of the stand mixer should feel hot to the touch once all the sugar syrup is in. Turn the mixer up to high speed and let it whip until the bowl feels room temperature again. This may take about 10 minutes.
Once the bowl feels cool, you can start adding your butter in small chunks. It is vital that the butter is softened, but not too warm. For best results, I like to take my butter out of the fridge about an hour before I start making the buttercream. Add the butter to the mixer one small chunk at a time, letting it whip until it is incorporated in between each addition.
As you are adding the butter the consistency of the frosting will change. At first, it will look soupy and runny, don't panic, just keep going. Eventually, the frosting will thicken up and look lovely and glossy. If it doesn't, turn the mixer up to maximum speed and whip for a further minute.
Once all the butter is added, add the vanilla and pistachio paste and whip on high speed for a minute. Do a taste test and decide if you'd like to add the salt. This is a personal preference, but I love it in there as once it's on the dacquoise cake it makes a gorgeous sweet and salty contrast.
Assemble the Pistachio Dacquoise Cake
Place one of the dacquoise cakes on your cake board or serving plate.
Put the pistachio french buttercream in a piping bag and snip the end off. Squeeze little blobs of buttercream around the outside of the dacquoise.
You can either add blobs all over the layer, or just do the outer ring and spread some frosting in the middle of the cake.
Place the other dacquoise layer gently on top of the buttercream and repeat the process, piping blobs of French buttercream all over the top of the cake.
Decorate with some chopped pistachios around the outside of the cake.
Notes
See the blog post above for more detailed information on each of the steps in the recipe, as well as information on ingredients, equipment, variations, and storage. If you can't find pistachio paste, it's super easy to make your own. See my blog post on how to make your own pistachio paste.If you don't have cake rings, you can just spread the dacquoise batter onto parchment paper in a circle shape. The dacquoise won't be quite as neat but it will still work. It's easiest to trace a 7-inch circle onto the parchment paper first so you have a guide to follow. *Cream of Tartar helps to stabilize the meringue but it is not necessary for the success of the recipe. If you don't have it you can substitute for twice as much lemon juice, or leave it out entirely. You'll just have to be extra careful not to knock too much air out of the meringue when folding.