Make the cake: Preheat the oven to 375 degrees F. Coat an 11-by-17-inch rimmed baking sheet or jelly roll pan with cooking spray; line the bottom with parchment paper and spray the parchment. Beat the egg whites and lemon juice in a large bowl with a mixer on medium speed until soft peaks form, about 3 minutes. Add 1/2 cup confectioners' sugar. Increase the speed to medium high and beat until stiff peaks form, 3 to 4 minutes.
Whisk the egg yolks with the vegetable oil, milk and vanilla seeds in a separate large bowl until smooth. Sift the remaining 1 cup confectioners? sugar, the flour, baking powder and salt into a medium bowl; whisk into the yolk mixture until smooth. Stir about one-third of the whites into the batter with a rubber spatula, then fold in the remaining whites in two batches. Spread the batter in the prepared pan; bake until the cake springs back when gently pressed, 12 to 14 minutes.
Immediately sift confectioners' sugar generously over the top of the warm cake and loosen the edges with a knife; lay a clean kitchen towel (not terry cloth) over the cake. Place another baking sheet over the towel and flip the cake with the baking sheets to invert it onto the towel. Remove the top baking sheet and parchment. Starting at a short end, roll up the cake and the towel together into a spiral. Transfer seam-side down to a rack and let cool completely.
Meanwhile, make the frosting: Whisk the egg whites, granulated sugar, vanilla seeds and salt in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water). Cook, whisking, until the sugar dissolves, 2 to 3 minutes. Transfer the mixture to a stand mixer and beat with the whisk attachment on medium-high speed until cooled and stiff peaks form, 6 to 8 minutes. Beat in the butter, 1 piece at a time, until combined (the mixture may look curdled). Add the lemon zest and juice. Switch to the paddle attachment and continue to beat until smooth, about 2 minutes.
Gently unroll the cake. Spread 2 cups of the frosting over the cake about 1/4 inch from the edges, then top with a layer of the lemon curd; swirl together. Reroll the cake (without the towel). Trim the ends with a serrated knife. Transfer to a platter and spread the remaining frosting on the top and sides of the cake.