To make the filling, beat the butter in a large bowl until light and fluffy. Add the powdered sugar, cocoa powder, salt, milk, and vanilla, and mix until well combined. Increase the mixer speed to high and beat until light and fluffy. Fold in the mascarpone using a rubber spatula. Cover and refrigerate until ready to use.
To make the cake, preheat the oven to 400. Spray a 13 x 18 baking sheet with cooking spray, then line with parchment paper. Spray the parchment paper with cooking spray and set aside.
In a small bowl, whisk together the cocoa powder, salt, and flour.
In a large bowl or the bowl of a stand mixer, beat the eggs and sugar together until they are light in color, fluffy, and slightly thickened. Add the vanilla and half of the cocoa powder mixture, and mix until combined. Add the remaining cocoa powder, and mix until combined.
Pour the cake batter into the prepared pan, spreading evenly. Tap lightly on the counter to remove any large air bubbles. Bake the cake 8-10 minutes, until a toothpick inserted in the middle comes out clean, and the edges of the cake pull away from the sides of the pan.
Dust a clean kitchen towel evenly with powdered sugar, then invert the cake onto the towel. Remove the parchment paper and sprinkle the cake with powdered sugar. Roll the cake up into a spiral, starting from the long edge, wrapping the towel inside the cake. Let cool 15-20 minutes.
After the cake has cooled, gently unroll the cake and spread the filling evenly across the cake, reserving about 1/2 cup of filling. Gently roll up the cake, being careful not to roll so tightly the filling squishes out. Wrap the cake in plastic wrap and refrigerate at least two hours.
After the cake has chilled, trim 1/2 inch from both ends of the log to display the spiral pattern. Make a diagonal cut about 3 inches from one end of the cake, then attach the cut piece on one side of the log with the remaining filling. Fill in any gaps with the filling.
To make the ganache, heat the heavy cream to boiling in a microwave safe bowl or measuring cup, about 2 minutes. Place the chocolate chips in a medium bowl. Pour the cream over the chocolate chips and let sit for one minute, then stir until the ganache is smooth and the chocolate chips are completely melted.
Let the ganache cool 10-15 minutes, until slightly thickened. Spoon the ganache over the yule log, covering all surfaces except the cut ends of the cake. Refrigerate 15 minutes, then make a swirl pattern in the ganache using the back of a spoon to resemble tree bark. Refrigerate at least one hour, until set. When ready to serve, move the cake to a serving platter and sprinkle with powdered sugar.