Preheat the oven to 350 degrees F and line a 10x15 inch sheet pan with parchment paper that sticks up about an inch on all sides for easy removal later. Spray the parchment paper with oil.
In a large bowl, whisk the flour, sugar, cocoa powder, baking powder, baking soda and salt. Now add the milk, vinegar, oil, applesauce and vanilla and mix with an electric mixer (or just a whisk) until combined.
Pour into the prepared pan and bake for 10-12 minutes, or until a toothpick inserted comes out clean.
While the cake is baking, lay a large, thin, clean dish towel on the counter and dust with a few tablespoons of cocoa powder, all over the towel. This will help the cake not to stick to the towel.
When the cake is done baking, remove from the oven and, using oven mitts, carefully and swiftly invert the cake onto the dish towel. Peel off the parchment paper that was on the bottom of the cake.
Starting with the shorter end of the cake, begin tightly rolling the hot cake in the towel. Do this process slowly and gently. Let the cake cool completely rolled up in the towel at room temperature. This will take about 3 hours, so plan ahead.