With the rack in the middle position, preheat the oven to 180 °C (350 °F). Butter the sides of the baking sheet.
In a bowl, whisk the eggs, sugar and vanilla with an electric mixer for 5 minutes.
In another bowl, sift the flour and baking powder. With a spatula, fold into the previous mixture. Fold in the pistachios and butter. Spread the batter on the baking sheet. Bake for 13 minutes or until a toothpick inserted in the centre of the cake comes out clean.
Right out of the oven, flip the cake onto a parchment paper sprinkled with sugar. Remove the baking parchment paper. Roll the cake onto itself and let cool to room temperature.
. Buttercream : Place the egg yolks in large bowl. Set aside.
In a saucepan, bring the sugar and water to a boil. Simmer until a candy thermometer reads 105 °C (220 °F) and the mixture forms a thick syrup, about 10 to 12 minutes.
Pour the hot syrup in a stream over the egg yolks while beating with an electric mixer. Beat until completely cooled, about 15 minutes. Add the butter, a few pieces at a time, continuing to beat until the mixture is very smooth. Add the vanilla.
Unroll the cake and cover it with buttercream. Roll back again. Refrigerate for about twenty minutes and serve.
