1. Preheat your oven to 400°f and spray a half sheet pan with vegetable oil and then line with parchment paper.
2. Place eggs sugar and salt in a mixer bowl over simmering water, whisking constantly until mixture reaches 110 degrees.
3. Quickly move bowl to stand mixer and whip at high speed until egg mixture is lightened, fluffy and cool.
4. Sift flour, cornstarch, cocoa together into one bowl. Stir to combine and then fold dry ingredients into egg mixture carefully trying not to deflate the air. Finally, fold in the melted butter until combined.
5. Pour batter into prepared pan and, using an offset spatula, gently spread the mixture into the sides of the pan to form an even layer.
6. Place pan in preheated oven immediately and bake for 10-15 minutes, or until cake is dry to the touch.
7. In the meantime, prepare a clean dish towel by sifting a thin layer of powdered sugar and cocoa onto it. Turn out the cake from the pan onto the dishtowel and very carefully remove the parchment paper from the top. Starting from the shorter end of the cake, begin rolling the cake up onto itself, bringing the dish towel with it, until the cake is completely rolled up. Allow to cool completely.
8. Meanwhile, make the filling. In a stand mixer, whip together cream, mascarpone, vanilla, spices and vanilla until thickened.
9. Once cake is cool, unroll it and use an offset spatula to spread the whipped cream all over the surface, leaving a very small border around the edges of the cake unfrosted. Gently roll the cake back up just like before (minus the towel), with the seam side facing down.
10. Take a large serrated knife and cut a small piece off from each end of the cake to create a clean looking edge. Then on one end, cut off a 2-inch slice, cutting at an angle so the larger side is 2 inches and the shorter side is 1 inch. Take this slice and place the straight side of it against the main cake to create a log effect.
11. Finally, make the frosting. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and salt on medium speed until fluffy, 1 to 2 minutes. Gradually add the confectioners’ sugar and beat until smooth, 2 to 3 minutes. Add the vanilla, melted chocolate and cinnamon and beat until just incorporated, being careful not to incorporate too much air into the frosting. Add milk 1 teaspoon at a time to achieve a spreadable consistency.
12. Use an offset spatula to frost the cake log and its side stump. Frost in large, rough swaths to achieve texture. Or if desired, use a fork to rake through the frosting for extra texture. Adorn the cake with cranberries and rosemary for a festive look.