Preheat your oven to 350°F (175°C). Grease a 15x10-inch jelly roll pan and line it with parchment paper.

In a large bowl, combine the flour, sugar, cocoa powder, baking powder, and salt. Whisk until well blended.

In a separate bowl, beat the eggs, milk, and vegetable oil until fully combined. Slowly add this mixture to the dry ingredients, stirring until just incorporated.

Pour the batter into the prepared pan and spread evenly. Bake for 12-15 minutes until the cake springs back when lightly touched.

While the cake is baking, prepare a clean kitchen towel by dusting it with powdered sugar. Once the cake is done, immediately turn it out onto the towel and remove the parchment.

Roll the cake tightly in the towel, starting from one short end. Allow it to cool completely while rolled.

For the filling, whip the heavy cream with powdered sugar and vanilla extract until soft peaks form.

Once the cake is cool, unroll it gently and spread the whipped cream filling over the cake, leaving a small border.

Carefully re-roll the cake (without the towel) and place it seam-side down on a serving platter.

To make the frosting, beat the softened butter, cocoa powder, and powdered sugar together until smooth. Add heavy cream until desired consistency is achieved.

Frost the outside of the cake with the chocolate frosting. Use a fork to create bark-like texture for a true log effect.

Chill in the refrigerator for at least 30 minutes before serving to set the frosting.