Preheat oven to 350°F. Grease a 10-inch x 15-inch jelly roll pan with butter. Line with parchment paper, grease the paper with more butter, then set aside.
In a large bowl, whisk together flour, cocoa powder, baking powder and salt. Set aside.
In a stand mixer fitted with a whisk attachment, whip eggs on medium-low speed until foamy, about 1 minute. Slowly add granulated sugar and brown sugar. Increase speed to high and beat until mixture has tripled in volume and is very pale, about 5 minutes. Beat in red food coloring. Mix in milk, melted butter, vinegar and vanilla. Remove bowl from mixer and use a rubber spatula to gently fold in dry ingredients until just combined and no traces of flour remain.
Pour batter into prepared pan and spread into an even layer. Bake until cake is firm to the touch and the center is springy, 12-15 minutes. Meanwhile, sift a generous layer of powdered sugar on a clean dish towel, at least 12×17-inch large. Remove cake from oven and run a knife around the edge of the pan to loosen it. Quickly and carefully invert it onto the prepared dish towel. Peel off the parchment paper. Gently roll the cake up with the towel, starting with the short edge. Let it cool completely in the towel.
In the bowl of a mixer fitted with a paddle attachment, combine cream cheese, butter and salt. Beat on medium speed until light and creamy, about 2 minutes. Add powdered sugar and vanilla and beat until smooth.
If refrigerated, let the cake warm up before unrolling to prevent cracking. Carefully unroll cake and spread cream cheese filling over the surface, leaving 1/2-inch border around the cake. Gently reroll the cake over the filling, without the towel, to create a tight log. Place it seam side down on a clean baking sheet, cover loosely with plastic wrap and refrigerate at least 1 hour. To serve, dust with more powdered sugar.