Preheat your oven to 375°F. Grease a 10x15 inch jelly roll pan, line with parchment paper, and grease the parchment. Separate the eggs carefully, ensuring no yolk contaminates the whites. Sift the flour and cornstarch together into a bowl. Measure and prepare all remaining ingredients, and place the heavy whipping cream in the refrigerator to chill.

In a large bowl, whisk the room temperature egg yolks with 1/2 cup sugar for 2-3 minutes until pale yellow and slightly thickened. Gradually drizzle in the vegetable oil while whisking constantly to form a smooth emulsion. Whisk in the milk and 5 teaspoons vanilla until combined. Using a spatula, gently fold in the sifted flour and cornstarch mixture just until no flour streaks remain, being careful not to overmix.

In a clean, grease-free bowl, beat the 12 egg whites with the vinegar using an electric mixer on medium-high speed for 3-4 minutes until soft peaks form. Gradually add 1/2 cup sugar while continuing to beat until stiff, glossy peaks form, about 2-3 minutes more. Fold the meringue into the yolk batter in two additions using a gentle folding motion with a spatula to preserve the airiness.

Transfer about 1/2 cup of the batter to a small bowl and stir in 1 teaspoon food dye until evenly colored. Transfer the tinted batter to a piping bag fitted with a round tip or a squeeze bottle. Pipe the red flag stripes and design onto the prepared parchment in the pan. Immediately place the pan in the freezer for exactly 5 minutes to set the design.

Remove the pan from the freezer and carefully pour the remaining batter over the frozen piped design. Use an offset spatula to spread the batter evenly without disturbing the pattern underneath. Bake for 8-10 minutes until the top is light golden and springs back when gently pressed. The cake should be tender and slightly underbaked.

While the cake is still hot, place a damp kitchen cloth (about 16x20 inches) on your work surface. Invert the hot cake onto the cloth and carefully peel away the parchment paper so the flag design is now on the bottom. Starting from the short end, roll the cake tightly in the damp cloth and let it cool on the counter for 1 hour at room temperature.

While the cake cools, prepare the filling. In a bowl, combine the chilled heavy whipping cream, 6 tablespoons sifted powdered sugar, 1/4 teaspoon salt, 1 1/2 teaspoons vanilla, and 1/2 teaspoon lemon zest. Beat with an electric mixer on medium-high speed for 2-3 minutes until stiff peaks form, being careful not to overbeat.

Once the cake has cooled completely, gently unroll it from the cloth onto your work surface with the flag design facing up. Spread the whipped cream filling evenly over the cake, leaving a 1/2-inch border around the edges. Distribute the sliced strawberries evenly over the whipped cream. Tightly roll the cake back up without the cloth. Transfer to a serving platter, cover, and refrigerate for 15-20 minutes to set before slicing.