For the onions and mushrooms: Preheat the oven to 400 degrees F.
Heat 2 tablespoons butter in a medium skillet over medium heat, then add the onions and garlic and sweat for 4 to 5 minutes. Add the remaining 2 tablespoons butter to a second medium skillet over medium-low heat, then add the mushrooms. Cook slowly until golden, 8 to 10 minutes. Add the onions to the mushrooms.
For the steaks: Heat a 10-inch cast-iron pan over high heat for 2 minutes. Add the olive oil. Sprinkle the steaks heavily with sea salt and pepper. When the oil begins to shimmer, add the steaks and saute until rare (internal temperature of 135 degrees F), 3 minutes per side. Remove and set aside. Turn the heat to medium and add the garlic and spinach to the pan. Saute, stirring occasionally, until the spinach is just wilted. Remove to a cutting board and chop. Add the spinach to the onions and mushrooms. Turn heat to low to keep warm.
For the eggs: Whisk the eggs and heavy cream together in a medium bowl until very smooth. Melt the butter in a small, heavy-bottomed saucepan. Add the eggs and cook, stirring constantly using a wooden spoon, until they reach a soft, custard-like consistency, 5 to 7 minutes.
For the lettuce: Heat a wok or large skillet over medium heat. Add the olive oil. When smoking slightly, add the lettuces and cook, tossing, until wilted, 1 to 2 minutes. Add the mushroom mixture to the wok and toss. Turn the heat to low and gently mix in the scrambled eggs.
Put the lettuce, egg and mushroom mixture on a large serving platter. Slice the steaks and arrange around the vegetables. Place the Shoestring Potatoes on top and serve.
Peel the potatoes and julienne into thin matchsticks using a mandoline. Fill a large bowl with cold water and soak the potatoes for 1 hour. Drain and dry the potatoes in a salad spinner.
Heat the oil to 325 degrees F in a Dutch oven. Fry the potatoes in batches until cooked through, 3 to 4 minutes. Let drain on paper towels, 5 to 10 minutes. Refry until golden brown, another 3 to 4 minutes. Remove with a slotted spoon and drain on paper towels. Sprinkle with sea salt.