1. Fill a large pot with water. Add potatoes and one tablespoon of salt. Bring to a boil and lower the heat to a simmer. Cook until the potatoes are very soft.
2. Meanwhile, heat three tablespoons oil in a large frying pan. Add onion and sauté until soft and golden, about 15–20 minutes. Remove half the onions from the pan and set aside. Add mushrooms to the pan and sauté over high heat, stirring every few minutes. (The mushrooms will release a lot of liquid. Keep the heat on high to evaporate it quickly.) Cook until mushrooms are soft and juices have cooked out. Season with salt and pepper to taste. Set aside.
3. When the potatoes are soft, drain and mash well. While they’re still hot, add the onions that you set aside, along with the margarine and oil, and mix well. Allow to cool slightly and add the beaten eggs, mixing quickly so the eggs don’t cook. Add flour and mix until well incorporated. Season lightly with some additional salt and pepper to taste.
4. Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius). Spray a 10-inch (26-centimeter) bundt pan very generously with cooking spray. Sprinkle all sides of the pan with everything spice, trying to coat it as evenly as possible. It should stick to the cooking spray in most areas, but it’s okay if there are some empty spots.
5. Pat half the potato mixture into the pan. Run a large soup spoon around the center of the mixture, pressing down slightly to create a tunnel. Spoon the mushrooms and onions into the tunnel, distributing evenly around the pan. Try to keep the filling inside the indentation. Cover with the remaining potato mixture, spooning it gently over the filling and patting it down. Smooth out the top of the mixture and brush with beaten egg. Sprinkle generously with more everything spice.
6. Place pan into the oven and bake for 55–60 minutes or until puffed and golden. Remove from oven and allow to cool for about 10 minutes. Run a knife around the edges of the pan to loosen and invert onto a wire rack. Allow to cool an additional 10 minutes before slicing.