Heat your oven to 375°F and grease a 10x15 inch jelly roll pan, then line it with parchment paper and grease the parchment as well. Separate your eggs carefully, ensuring no yolk gets into the whites—even a tiny bit of yolk will prevent the whites from whipping properly. Measure out all your dry ingredients (flour and cornstarch) and sift them together into a bowl to remove lumps. Have your milk, vanilla, vinegar, lemon zest, and food dye measured and ready. Chill your heavy whipping cream in the refrigerator while you work on the cake.

Whisk together the room temperature egg yolks with 1/2 cup sugar until the mixture is pale yellow and slightly thickened, about 2-3 minutes—this creaming step helps incorporate air and creates a lighter cake. Slowly drizzle in the vegetable oil while whisking continuously to create an emulsion. Add the milk and 5 teaspoons vanilla and whisk until combined. Gently fold in the flour and cornstarch mixture from Step 1 using a spatula, being careful not to overmix—just combine until no flour streaks remain.

In a separate clean bowl (even a tiny bit of oil will prevent proper whipping), beat the 12 egg whites with the vinegar using an electric mixer on medium-high speed for about 3-4 minutes until soft peaks form. Gradually add 1/2 cup sugar while continuing to beat until stiff peaks form and the mixture is glossy, about another 2-3 minutes total. Fold this meringue into the yolk batter from Step 2 in two additions, gently folding with a spatula rather than stirring to maintain the airiness—I like to fold just until a few white streaks remain with the first addition, then complete the folding with the second, as this prevents deflating the meringue.

Transfer about 1/2 cup of the batter to a small bowl and add 1 teaspoon of food dye (adjust the amount based on how intense you want the color), stirring until evenly tinted for your red stripes. Transfer this colored batter to a piping bag fitted with a round tip (or use a squeeze bottle for easier control). Pipe the red stripes and flag design onto the prepared parchment paper in the prepared pan. Immediately place the pan in the freezer for exactly 5 minutes—this sets the design so it doesn't blend into the base cake when you pour the remaining batter over it.

Remove the pan from the freezer and carefully pour the remaining cake batter from Step 3 over the frozen design, using an offset spatula to spread it evenly without disturbing the piped pattern below. Bake for 8-10 minutes until the top is light golden and springs back when lightly touched with your finger—the cake should still be very tender and slightly underbaked because it will continue cooking slightly as it cools and sits on the warm pan. The cake is done when a toothpick inserted in the center comes out mostly clean with just a few moist crumbs.

Working quickly while the cake is still hot, place a damp kitchen cloth (about 16x20 inches) on your work surface and gently turn the hot cake out onto it. Carefully peel off the parchment paper—the design should now be on the bottom of the cake facing down. Starting from the short end, tightly roll the cake up in the damp cloth and let it cool on the counter for 1 hour. The damp cloth helps the cake retain moisture and makes it easier to roll without cracking. I let it cool completely at room temperature because this helps the cake set properly before filling.

While the cake cools, prepare the whipped cream filling by combining 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 in a bowl. Beat with an electric mixer on medium-high speed for about 2-3 minutes until stiff peaks form—be careful not to overmix or you'll get butter. Once the cake has cooled, unroll it gently from the cloth onto your work surface (the flag design is now on top). Spread the whipped cream evenly over the interior of the cake, leaving a 1/2-inch border around the edges. Distribute the sliced strawberries evenly over the cream, then tightly roll the cake back up without the cloth. Transfer to a serving platter, cover, and refrigerate for 15-20 minutes before slicing to help it set.