Make the Pistachio Dacquoise:

1. Prepare the oven: Preheat to 465°F (240°C). Line a baking sheet with parchment paper and place two 7-inch pastry rings on it, ungreased.
2. Grind dry ingredients: Process pistachios in a food processor until finely ground but not oily. Add powdered sugar and cake flour; pulse to combine.
3. Whip egg whites: Beat egg whites with cream of tartar and salt until frothy. Gradually add sugar and whip to stiff peaks.
4. Combine: Gently fold the pistachio mixture into the whipped egg whites.
5. Bake: Divide batter between the pastry rings and bake at 350°F (180°C) for 15 minutes. Cool in the oven with the door slightly open.

Make the French Buttercream:

1. Prepare egg yolks: Whip yolks until pale and thick.
2. Cook sugar syrup: Heat sugar and water to 240°F (115°C). Gradually pour into the egg yolks while mixing continuously.
3. Add butter and flavor: Mix in softened butter, a little at a time, until smooth. Add vanilla, pistachio paste, and salt to taste.

Assemble the Cake:

1. First layer: Place one dacquoise layer on a cake board and pipe buttercream around the edges.
2. Fill and stack: Fill the center with buttercream and place the second dacquoise layer on top.
3. Decorate: Frost the top with buttercream and garnish with chopped pistachios.