Preheat your oven to 375°F (190°C). Grease a 10x15 inch jelly roll pan. Line it with parchment paper, leaving a small overhang on the short sides to act as handles. Grease and lightly flour the parchment paper.

In a medium bowl, sift together the all-purpose flour, cocoa powder, baking powder, and salt. Set aside.

In a large bowl, use an electric mixer to beat the egg yolks with 1/4 cup of the granulated sugar on high speed until the mixture is pale yellow, thick, and ribbon-like (about 3-4 minutes). Beat in the vegetable oil, hot water, and vanilla extract.

Gently fold the sifted dry ingredients into the egg yolk mixture until just combined. Do not overmix.

In a separate, clean, and dry bowl, beat the egg whites on medium speed until they become foamy. Gradually add the remaining 1/4 cup of granulated sugar and increase the speed to high. Beat until stiff, glossy peaks form.

Gently fold about one-third of the beaten egg whites into the chocolate batter to lighten it. Then, carefully fold in the remaining egg whites in two additions until no white streaks remain.

Pour the batter into your prepared pan and spread it into an even layer with an offset spatula. Bake for 10-12 minutes, or until the top of the cake springs back when lightly touched. Be careful not to overbake.

While the cake bakes, lay a clean kitchen towel (not terry cloth) on your work surface and dust it generously with cocoa powder or powdered sugar. As soon as the cake comes out of the oven, immediately invert it onto the prepared towel. Carefully peel off the parchment paper.

Starting from one of the short ends, gently and tightly roll the warm cake up with the towel inside. Place the rolled cake, seam-side down, on a wire rack to cool completely (about 1 hour).

In a chilled bowl, use an electric mixer to whip the very cold heavy cream, sifted powdered sugar, and vanilla extract until stiff peaks form.

Once the cake is completely cool, carefully unroll it. Spread the whipped cream filling evenly over the surface, leaving a 1-inch border on all sides.

Gently re-roll the cake, this time without the towel. Wrap the cake roll tightly in plastic wrap and refrigerate for at least 1 hour to set the filling.

Place the finely chopped semi-sweet chocolate in a medium heat-proof bowl.

Heat the 1/2 cup of heavy cream in a small saucepan over medium heat until it just begins to simmer around the edges. Immediately pour the hot cream over the chopped chocolate. Let it sit undisturbed for 5 minutes to melt the chocolate.

Whisk the mixture gently, starting from the center and working your way out, until the ganache is completely smooth and glossy. Let it cool at room temperature for about 20-30 minutes, or until it has thickened to a spreadable consistency.

Unwrap the chilled cake roll and place it on a wire rack set over a baking sheet (to catch drips). Pour the thickened ganache over the cake, using an offset spatula to spread it evenly over the top and sides. You can trim the ends of the roll for a cleaner look before applying the ganache.

Refrigerate the cake for another 30 minutes to allow the ganache to set firmly before slicing with a hot knife and serving.