Preheat the oven to 325 degrees F (165 degrees C). Spray a jelly roll or sheet pan with cooking spray and line with parchment paper.

Sift flour, cocoa, salt, and baking soda together in a medium bowl.

Beat eggs and sugar in another bowl with an electric mixer on high speed until tripled in volume, about 5 minutes. Add 1/2 teaspoon vanilla extract. Beat in dry ingredients on low speed in thirds, alternating with melted butter, until just combined. Pour into the prepared pan (the thickness should be less than 1/4 inch).

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 8 to 9 minutes. Allow to cool completely.

Flip cake upside-down out of the pan onto a new sheet of parchment paper. Remove the used parchment paper from the top of the cake gently. Roll cake, starting at one end of the paper, into a tube until it is completely rolled into the parchment. Refrigerate for 2 hours.

Whip cream, powdered sugar, and vanilla extract together in a bowl. Mix in chocolate-hazelnut spread. Roll out the cake and spread filling evenly. Roll up the cake with the filling, keeping in mind that the cake might crack or tear easily.