For the cake: Position a rack in the center of the oven and preheat to 375 degrees F. Line 13-by-18-by-1-inch rimmed baking sheet with parchment so that 2-inches overhang the long sides. Generously grease the parchment and unlined parts of the baking sheet with butter. Use a fine sieve to generously dust a large, clean kitchen towel confectioners' sugar until it is completely covered (about 1/2 cup). Set aside.
Beat the eggs in large bowl with an electric mixer on high speed until very thick and pale yellow, about 5 minutes. Gradually beat in the granulated sugar and continue to beat until thick and the sugar has almost dissolved, 3 minutes. Beat in 2/3 cup water and the vanilla. Gradually add flour, beating just until the batter is smooth and no lumps remain. Pour into the prepared baking sheet and spread to the corners. Bake until the top is golden and a toothpick inserted in the center comes out clean, 15 to 18 minutes. Immediately loosen the cake from sides of baking sheet using a paring knife and invert the cake onto prepared towel. Carefully peel off the parchment. Using a sharp Chef's or serrated knife, cut the hot cake crosswise into four 4-inch-wide strips. Flip each over so that the top of the cake is facing up. Push the strips together so that the cut sides are touching each other and the cake looks whole and uncut. Position the cake so that the long end is facing you. Using the towel to help you, carefully roll cake and towel, forming a tight log. Let log cool on rack for 3 minutes. This will allow the cake to roll without cracking or breaking. Carefully unroll the cake, and let sit on rack until completely cooled. The ends will be curled up and there might be surface cracks on the top of the cake and that's OK.
For the frosting: Beat together the butter and cream cheese in large bowl with an electric mixer on high speed until light and fluffy, 3 to 4 minutes. Reserve 1/4 cup brown sugar and set aside. While beating on high, gradually add the remaining brown sugar (making sure to break up any clumps before you put it in the mixer), vanilla, cornstarch, 2 tablespoons cinnamon and salt until sugar is almost dissolved and no lumps remain, 3 to 5 minutes. If the frosting looks loose, chill for 30 minutes and continue to beat until thick and smooth. Frost the cake with half of the frosting, filling in any cracks and smoothing out the surface.
Remove cake strips from towel and arrange on a clean work surface so that the strips are touching (short-end to short-end) to form a long cake strip running the length of your counter. Starting at one end, carefully roll-up 1 of the short ends tight enough so the cake sticks to the frosting but not so tight that the frosting gets squeezed out. Continue rolling-up each successive strip to form a fat roll. Carefully transfer to a cake plate or stand. Press on the sides of the cake to form an even and circular shape. Frost the sides of the cake with 1 cup of the remaining frosting. Refrigerate until chilled, about 30 minutes. Frost the sides of the cake again with the remaining frosting making sure it's about as thick as the inside layer of frosting.
Mix together the reserved 1/4 cup brown sugar and 1 tablespoon cinnamon in small bowl. Pass through a course sieve to break up any lumps. Lightly sprinkle on top of the cake, making the frosting look like the cinnamon part of the cinnamon roll. Chill, uncovered for at least 1 hour and up to overnight.
When ready to serve, stir powdered sugar and cream together in a small bowl until smooth and creamy. The mixture should be thick but pourable, add another tablespoon of cream if mixture is too thick. Drizzle over chilled cake so that some of the cream mixture drips down the sides (but don't cover up too much of the top of the cake or you won't see the cinnamon swirl!)
When measuring flour, we spoon into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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