Beat the eggs with the sugar and salt using a hand mixer until the mixture is pale, thick, and very foamy.
Gently fold in the flour, baking powder, and vanilla using a spatula, mixing just until you get a smooth batter with no dry spots.
Peel the apples, then grate them on the large holes of a box grater.
Line a 16×24‑inch (40×60 cm) rimmed baking sheet with parchment paper, lightly grease it with a bit of butter, then spread the grated apples in an even layer to cover the entire surface.
Pour the batter evenly over the grated apples and smooth the top, then bake at 350°F (180°C) for 15–20 minutes, until the top is lightly golden and springs back to the touch.
When the baking time is up, carefully invert the hot cake onto a sheet of parchment on the work surface, peel off the parchment it baked on, then use the clean parchment underneath to gently roll it up into a log while still warm.
Dust the outside of the roll generously with powdered sugar, let it cool completely, then slice and serve.