Shred the cabbage, add salt, and let it sit for 10 minutes to release excess water.
In a large skillet, heat the butter or oil over medium heat. Add the sugar and caramelize until golden brown.
Squeeze the cabbage to remove excess water. Add the cabbage to the caramelized sugar and cook over medium heat until softened, stirring occasionally. Season with black pepper.
Let the cabbage cool completely. Lay out the strudel dough sheets on a clean kitchen towel and brush each sheet with melted butter.
Layer 2-3 sheets on top of each other to strengthen the strudel. Evenly spread the cooked cabbage filling over the top layer.
Using the kitchen towel, carefully roll up the strudel and place it on a baking sheet lined with parchment paper.
Brush the top of the strudel with beaten egg to achieve a beautiful golden-brown color during baking.
Preheat the oven to 350°F (180°C) and bake for 30-35 minutes, or until the strudel is golden brown and crispy.
Let it cool slightly before slicing and serving.
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