Preheat oven to 350°F (175°C). Line a 15x10-inch jelly roll pan with parchment paper, letting it go up the sides slightly, and lightly grease the parchment.
Separate the eggs. Beat the yolks with 1/2 cup sugar until thick and pale yellow, about 3-4 minutes. Add cocoa powder, vanilla, and salt; beat until smooth.
In a clean bowl, beat the egg whites until soft peaks form. Gradually add the remaining 1/4 cup sugar and beat until stiff peaks form.
Gently fold one-third of the egg whites into the chocolate mixture to lighten it, then carefully fold in the remaining egg whites. Pour the batter into the prepared pan and spread evenly.
Bake for 10-12 minutes, until the cake springs back when lightly touched. Do not overbake. Meanwhile, lay a clean thin kitchen towel on the counter and dust generously with powdered sugar.
Immediately run a knife around the edges of the hot cake, invert it onto the prepared towel, and peel off the parchment paper. Roll the cake up with the towel inside, starting from a short end.
Place the rolled cake seam-side down on a cooling rack and let it cool completely, about 1 hour.
In a large chilled bowl, beat the heavy cream until it starts to thicken. Add powdered sugar and vanilla, and continue beating until stiff peaks form. Cover and refrigerate.
Beat softened butter until creamy. Sift together powdered sugar and cocoa powder. Gradually add the sugar mixture to the butter, alternating with heavy cream, then add vanilla. Beat on medium-high for 3-4 minutes until light and fluffy.
Carefully unroll the cooled cake. Spread the whipped cream filling evenly over it, leaving a 1/2-inch border. Gently roll the cake back up without the towel and place seam-side down on a serving platter.
If desired, cut a 2-inch piece off one end at a diagonal and attach it to the side of the log with buttercream to create a "branch."
Spread chocolate buttercream all over the log. Use a fork or offset spatula to drag lengthwise through the frosting to create bark-like texture. Create swirls on the ends for tree rings.
Dust with powdered sugar to look like snow and decorate with rosemary sprigs, sugared cranberries, or meringue mushrooms if desired. Refrigerate for at least 1 hour before serving.