Preheat the oven to 400 degrees F (200 degrees C). Line a jellyroll pan with parchment paper.
Beat eggs in a bowl with an electric mixer. Add salt and continue beating until eggs are light and fluffy, about 5 minutes. Add 1/2 cup sugar, 1 tablespoon at a time, and beat for an additional 5 minutes. Stir in oil and vanilla extract.
Mix flour and baking powder together. Sift into the egg mixture a little at a time, folding in with a spatula. Spread batter onto the prepared baking sheet.
Bake in the preheated oven until cake is lightly browned, 10 to 12 minutes.
Lay a clean dish towel on a flat work surface and sprinkle dish towel with 2 teaspoons sugar. Invert sponge cake onto the dish towel and remove parchment paper. Roll up cake with the dish towel and allow to cool completely, about 1 hour.
Carefully unroll sponge cake. Spread chocolate-hazelnut spread on top, leaving a little room on the edges. Carefully roll up and wrap with plastic wrap. Chill in the refrigerator for 2 hours. Cut into slices.