With the rack in the middle position, preheat the oven to 190 °C (375 °F). Butter and line a 43 x 30-cm (17 x 12-inch) baking sheet with parchment paper, letting it hang over two sides.
In a bowl, combine the flour, baking powder, and salt. Set aside.
In another bowl, beat the eggs and sugar with an electric mixer until the mixture triples in volume and is thick enough to form a ribbon when the beater is lifted from the bowl, about 10 minutes. Add the oil and vanilla.
Sift the dry ingredients over the egg mixture, gently folding as you go, with a spatula or whisk.
Spread on the baking sheet and bake for 12 to 15 minutes or until a toothpick inserted in the centre of the cake comes out clean. Let cool for 5 minutes.
Unmould the cake onto parchment paper. Remove the paper from the top. Roll the warm cake with the paper from the shorter side. Let cool completely.
In a heavy saucepan, bring all the ingredients to a boil, except for the butter, stirring to dissolve the sugar. Attach a candy thermometer in the centre of the pan and simmer, without stirring, until the thermometer reads 112 °C (234 °F). Remove from the heat and place the pan in an ice water bath. Let cool, without stirring, until the thermometer reads 43 to 50 °C (110 to 122 °F) or 20 to 30 minutes.
Remove the pan from the water. With an electric mixer, beat the mixture until it loses its gloss but is still soft, about 2 minutes.
Beware, the next step must be done quickly. Spread three-quarters of the sugar fudge on the unrolled cake. Roll again. Set aside.
Continue to beat the remaining sugar fudge and add the butter, one or two cubes at a time, beating until smooth. If necessary, refrigerate for 5 to 10 minutes or until the buttercream is spreadable. Place the log on a serving dish and frost with the sugar fudge buttercream. Store at room temperature.
Beware; this dessert is for serious sweet tooth!
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.