Preheat your oven to 350°F (175°C). Grease a 10×15 inch jelly roll pan and line it with parchment paper, leaving a little overhang on the sides. Grease the parchment paper as well to ensure the cake releases easily.
In a large mixing bowl, beat the eggs and granulated sugar on high speed for about 5 minutes. The mixture should become thick, pale, and double in volume. This aeration is key to a light, flexible cake.
Gently whisk in the melted butter, vanilla extract, white vinegar, and red food coloring until the color is uniform and vibrant red.
Sift together the flour, cocoa powder, baking powder, and salt. Gradually fold the dry ingredients into the wet mixture using a spatula. Be careful not to overmix, as you want to keep the air in the batter.
Spread the batter evenly into the prepared pan. Bake for 12 to 15 minutes, or until the top of the cake springs back when lightly touched. Do not overbake, or the cake will become brittle and crack.
While the cake is baking, lay out a clean kitchen towel on the counter and dust it generously with powdered sugar. As soon as the cake comes out of the oven, turn it out onto the towel. Peel off the parchment paper. Starting from the short end, roll the warm cake and the towel together. Let it cool completely in this rolled shape.
In a medium bowl, beat the softened cream cheese and butter until smooth. Add the powdered sugar and vanilla extract, beating until the frosting is creamy and stiff enough to hold its shape.
Once the cake is completely cool, gently unroll it. Spread the cream cheese filling evenly over the surface, leaving a small border at the edges. Roll the cake back up tightly (without the towel this time). Wrap in plastic wrap and refrigerate for at least 1 hour before serving.