Preheat the oven to 375 degrees F (190 degrees C). Line the bottom and sides of a 15 1/2×10 1/2×1-inch jelly roll pan with waxed paper.
Beat egg whites, vanilla extract, and salt using an electric mixer in a mixing bowl until soft peaks form. Beat in 1/2 cup white sugar gradually until combined.
Beat egg yolks in a separate bowl using an electric mixer until thick and lemon-colored. Fold into egg white mixture. Fold in flour and chopped walnuts carefully until combined. Spread batter into the prepared jelly roll pan.
Bake in the preheated oven until cake springs back when lightly touched and a toothpick inserted into the center comes out clean, about 12 minutes.
Remove from the oven and let cool for 5 minutes. Loosen sides of cake; turn out onto a towel sprinkled with sifted confectioners’ sugar. Peel off wax paper and let cool to lukewarm, 5 to 10 minutes.
Roll cake and towel together starting at the narrow end. Let cool completely on a wire rack, 15 to 20 minutes.
While cake cools, beat cold heavy cream and 1 tablespoon white sugar together in a mixing bowl using an electric mixer just until stiff peaks form.
Unroll cake and spread with whipped cream, reserving some for topping. Re-roll cake and let chill in the refrigerator for at least 30 minutes. Top with dollops of whipped cream and walnut halves. Slice and serve.
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