Preheat your oven to 180°C (350°F).
In a large skillet, heat the vegetable oil over medium heat. Add the chopped onion and sauté until translucent.
Add the shredded cabbage to the skillet, season with salt and pepper, and cook until the cabbage is softened and reduced in volume, about 10-15 minutes.
On a clean surface, lay out one sheet of phyllo pastry. Lightly brush it with vegetable oil, then fold it in half.
Place a portion of the cabbage filling along the long edge of the folded pastry. Roll it up tightly into a cylinder.
Repeat the process with the remaining pastry sheets and filling. Arrange the rolls in a lightly greased baking dish.
Brush the tops of the rolls with the beaten egg for a golden glaze.
Bake in the preheated oven for 20-25 minutes or until the pastry is golden brown and crisp.
Allow to cool slightly before serving. Enjoy your Banitsa with Cabbage warm.