Make the cake: Heat oven to 350°F and line the bottom and sides of a 10×15-inch jelly roll pan with a large piece of parchment (I used a pre-cut half-sheet rectangle). Dabbing the edges and corners of empty pan with a bit of water can help parchment stay in place. Coat the parchment with butter or nonstick spray.
Beat the eggs in a large bowl with a whisk or electric mixer, until well-mixed and bubbly. Add brown sugar, molasses, and applesauce, sour cream, or mascarpone and mix again. Sprinkle baking soda, salt, and spices over the batter and whisk thoroughly into batter, giving it several more whisks around the bowl than seems necessary. Sprinkle flour on batter and switch to a rubber scraper to stir. Stir slowly from the center of the batter, drawing in a little flour at a time until all the flour disappears. Scrape bowl well.
Pour batter into prepared pan and bake for 4 minutes; rotate pan 180 degrees. Bake for another 4 minutes, and check for doneness. The finished cake may look sticky and underbaked, but will not move when the pan is jiggled, and a tester inserted into the cake will come out clean or only with a couple sticky crumbs, not loose batter. Return to the oven for 2 minutes more, if needed, [updated] and in additional 2-minute increments until cake is set.
Transfer cake pan to a cooling rack and let cool for 5 minutes. While it cools, grab several things: A second large sheet of parchment paper, a large cutting board or flat tray, powdered sugar, and a mesh strainer to sift the sugar over the cake.
Run knife around between the cake edge and parchment, loosening it. Tilt the pan and gently tug the parchment and cake onto the cooling rack. Sift powdered sugar all over. Put the second sheet of parchment over the cake, and the cutting board over the parchment. With potholder-ed hands, grab cooling rack and board together, and flip cake onto the board. Gently, carefully peel back the parchment on the back of the cake. Sift powdered sugar all over the back of the cake.
Use the parchment underneath the cake to help you roll the cake from the short side into a snug coil and rest seam side down. Let it cool completely in this parchment log; this take a couple hours at room temperature or about an hour in the fridge. [But I chilled mine outside and it took about 45 minutes.]
Make the bark: Melt about 2/3 of the chocolate chips (you can eyeball it) in the microwave — give it 30 seconds, then stir, add another 30 seconds, if needed — or on the stove in a small pot over medium heat. Off the heat, stir in the remaining chocolate chips until melted. This will bring down the temperature of the chocolate so you can use it right away.
Spread chocolate thinly over a large piece of parchment paper — I use the same size I do for the baking pan, a 13×18-inch sheet. Roll it up into a log; I leave about 1 to 2 inches open in the center, for a looser coil — imagine rolling it around an invisible dowel. Chill in the fridge until firm [I sound like a broken record but I put mine on the patio for 15 minutes].
Make the cream: Place heavy cream, vanilla, and sugar (I use the lower amount; I prefer a barely sweetened cream here) in a large bowl and beat with an electric mixer — or with a large whisk — until the mixture holds stiff peaks. Whisk in mascarpone or sour cream until it disappears. This adds both a very slight tang and stabilization; the cream will stay thick for days.
Assemble the cake: Gently, carefully unroll your cooled cake. Spread it with about 2/3 of the whipped cream, in an even layer. Carefully re-roll your cake with the cream inside, carefully peeling off the parchment as you do. Sometimes, barely any cake stuck to the parchment. Other times, a bit more did and a scraper helped separate it. Place the rolled cake seam side down on the final serving platter.
Cover cake with remaining cream, leaving ends exposed. Slowly unroll your chocolate bark coil. The pieces of chocolate will separate in long and short curved pieces. Arrange them over the whipped cream to resemble bark. Shower cake with powdered sugar and decorate the tray with sugared cranberries.
Do ahead: Yule log will keep for several days in the fridge. The cake gets a bit softer every day.