Preheat your oven to 350ºF. In a large bowl, use an electric mixer to beat the eggs for about 5 minutes until they're frothy and pale yellow. Add the chocolate cake mix, water, and vegetable oil, then continue beating on low speed for 2 minutes until the batter is combined and smooth.
Line a 12x18-inch baking pan with parchment paper or a silicon liner. Lightly coat the liner and the sides of the baking sheet with baking spray for easy removal. Pour the cake batter from Step 1 onto the prepared pan and spread it evenly. Bake in the preheated oven for 12-15 minutes, or until a toothpick inserted in the center comes out clean. Once baked, let the cake cool in the pan for at least 5 minutes but no more than 10 minutes.
While the cake is still warm, lay out a clean linen dish towel and sprinkle it generously with powdered sugar on both sides to prevent sticking. Place the towel on top of the cake, then carefully flip the baking sheet so the cake lands on the towel. Gently tap to help release if needed. Peel off the parchment or silicone liner, then roll the cake up tightly with the towel still inside to help form its shape. Place the rolled cake seam-side down on a cooling rack and let it cool for at least 15 minutes.
Carefully unroll the cooled cake and evenly spread a layer of peppermint, vanilla, or marshmallow frosting (I prefer using peppermint for a festive touch) to the edges. Reroll the cake tightly to form a log and transfer it to a serving platter seam-side down. If desired, cut an angled slice off one end of the log to create a 'branch' and position it alongside the main cake for a traditional look. I always enjoy snacking on the leftover cake slice at this stage!
Frost the outside of the cake and the branch with the whipped chocolate frosting, covering the surface completely. Use a fork to gently drag along the frosting, creating a bark-like texture for a rustic, woodsy appearance. For a festive touch, garnish your Yule Log with fresh cranberries and rosemary sprigs to resemble holly and pine. Let your creativity inspire additional wintery decorations if you wish!