Preheat the oven to 350 degrees F. Brush an 11-by-17-inch jelly roll pan or rimmed baking sheet with vegetable oil and line with parchment paper; brush the paper with oil and dust with flour, tapping out the excess.
Whisk the flour, baking powder, baking soda and salt together in a medium bowl. Whisk the granulated sugar, vegetable oil, milk, eggs and vanilla together in a large bowl. Add the flour mixture in 2 batches, whisking until incorporated. Transfer half of the batter (about 1 1/2 cups) to a separate bowl. Tint one batch red and the other batch blue with the food coloring.
Spread the red and blue batters across the width of the prepared pan in alternating stripes, making each stripe about 2 inches wide (don't worry if the stripes aren't perfectly straight). Bake until the top springs back when pressed, 12 to 15 minutes. Let cool about 3 minutes in the pan. Dust with confectioners' sugar and spread a clean kitchen towel (not terry cloth) on top. Place a cooling rack on the towel, then invert the cake onto the rack; peel off the parchment. Roll up the cake and towel together, starting at a short end. Let cool completely on the rack, about 1 hour 30 minutes.
Gently unroll the cake (don't worry if the ends are still curled up). Spread the jam all the way to the edges. Reroll the cake (without the towel) and trim the ends. Top with dollops of whipped cream and garnish with shaved chocolate and mint. Slice and serve!