Bring 1 cup of sugar and 1 cup water to a simmer on the stove, stirring until sugar dissolves. Remove from heat and add cranberries. Let syrupy cranberries chill in fridge overnight or for an hour or two. Drain cranberries. Place remaining 1/3 cup sugar in a bowl and roll cranberries in it. Arrange them on a plate and chill until dry to the touch, about 45 minutes in the fridge.
Heat oven to 350°F and line the bottom and sides of a 10×15-inch jelly roll pan with parchment. 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. Sprinkle flour on batter and switch to a rubber scraper to stir. Stir slowly 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-minute increments until cake is set.
Transfer cake pan to a cooling rack and let cool for 5 minutes. Grab a second large sheet of parchment paper, a large cutting board or flat tray, powdered sugar, and a mesh strainer. Run a knife around the cake edge, 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 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.
Melt about 2/3 of the chocolate chips in the microwave — give it 30 seconds, then stir, add another 30 seconds, if needed — or on the stove over medium heat. Off the heat, stir in the remaining chocolate chips until melted. Spread chocolate thinly over a large piece of parchment paper and roll it up into a log. Chill in the fridge until firm.
Place heavy cream, vanilla, and sugar 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.
Gently 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, peeling off the parchment as you do. 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.