Preheat your oven to 350°F (175°C). In a large mixing bowl, sift together 1 cup of all-purpose flour, 3 tablespoons of cocoa powder, and 1 teaspoon of baking powder. In another bowl, whisk together 1 cup of sugar, 3 large eggs, 1 teaspoon of vanilla extract, and 1/4 cup of vegetable oil until smooth. Combine both mixtures, mixing until just fully combined and no lumps remain.

Line a 15x10 inch baking sheet with parchment paper and pour the batter evenly into the pan, spreading it out smoothly. Bake for 15-17 minutes, until the cake springs back when gently pressed and a toothpick comes out clean. Invert the cake onto a cocoa-dusted kitchen towel, rolling it up carefully while it's still warm. Let it cool for 2-2.5 hours.

In a separate bowl, whip 1 cup of heavy whipping cream with 1/2 cup of powdered sugar, 1 teaspoon of vanilla extract, and 2 tablespoons of cocoa powder until medium-stiff peaks form. Gently fold in 1 cup of mascarpone cheese until the mixture achieves a smooth, thick consistency.

Once the rolled cake has cooled completely, carefully unroll it on a flat surface, peeling away the towel. Spread the prepared filling evenly over the surface of the cake, ensuring to leave a small margin around the edges. Re-roll the cake tightly, starting from one end, and wrap it in plastic wrap. Chill in the refrigerator for about 30 minutes to set.

Prepare the ganache by heating 1 cup of heavy cream in a small saucepan until it just simmers. Pour it over 8 ounces of chopped chocolate in a bowl, letting it sit for a few minutes before stirring until smooth. Attach a cut piece of the rolled cake to the side with some ganache for a log effect. Coat the entire cake with ganache, then use a fork to create a bark-like texture.