1. Preheat oven to 350°. Line a 12- by 17-in. rimmed baking sheet with parchment paper. With a mixer on medium speed, beat egg yolks, 1 tsp. vanilla, the salt, and 2/3 cup sugar until thick and pale yellow, 3 minutes.
2. In a clean bowl, beat egg whites with clean beaters on high speed until soft peaks form, 1 to 1 1/2 minutes. Slowly rain in 1/3 cup sugar, beating until medium-stiff peaks form, 2 minutes.
3. Fold flour into yolk mixture just until incorporated. Fold in one-quarter of egg whites to lighten batter. Then gently fold in remaining egg whites just until no streaks remain. Spread evenly in prepared baking sheet.
4. Bake cake until it's very pale golden and slowly springs back when gently pressed in center, about 12 minutes. Loosen edge of hot cake with a thin knife, then invert cake onto a clean, smooth-textured kitchen towel laid on a flat surface. Remove parchment and let cake sit until just cool, 30 minutes.
5. Mix sliced berries with 4 tbsp. sugar in a bowl and set aside. With mixer, beat cream with remaining 3 tbsp. sugar and 1 tsp. vanilla until medium stiff.
6. Spread cake evenly with two-thirds of the whipped cream, then arrange sliced strawberries on top. Starting with a short side, roll cake up and set on a serving dish, seam side down. Frost with remaining cream and garnish with whole berries.