With the rack in the middle position, preheat the oven to 180 °C (350 °F). Line a 43 x 31-cm (17 x 12-inch) baking sheet with parchment paper. 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.
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.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.