Place cabbage in a large bowl with room for tossing. Sauté mushrooms in 1 Tbsp oil until nicely browned. Add salt. Carefully add balsamic vinegar (this works best without your face over the pan). Cook until reduced slightly, about 2 minutes. Add oregano. Remove and set aside. Add remaining ¼ cup olive oil to pan and cook onions to your desired finish. Add mushroom balsamic mixture and stir. Pour hot mixture over the cabbage. Toss until each shred is coated with dressing. Top with Parmesan (shave off curls with a vegetable peeler) and a poached egg.