Preheat the broiler. Put the potatoes in a medium saucepan and cover with water by 1 inch; season with salt. Bring to a boil and cook until the potatoes just start softening, about 5 minutes; drain and set aside. Season the steak with salt and pepper and set aside.
Meanwhile, heat 1 tablespoon olive oil in a large ovenproof skillet over medium-high heat. Add half of the shallot and cook, stirring, until slightly softened, about 2 minutes. Add the spinach and season with ½ teaspoon salt and a few of grinds of pepper. Cook, stirring, until the spinach is wilted and the liquid evaporates, about 3 minutes. Remove to a bowl; set aside.
Wipe out the skillet; add 1 tablespoon olive oil and heat over medium-high heat. Add the steak and cook until browned, about 4 minutes per side. Transfer to a plate and let rest.
Pour out any drippings from the skillet. Melt the butter in the skillet. Add the mushrooms and cook, stirring occasionally, until wilted, about 3 minutes. Stir in the remaining shallot, the potatoes, wine, ½ teaspoon salt and a few grinds of pepper. Drizzle with the remaining 1 tablespoon olive oil. Broil until browned, 8 minutes.
Stir the spinach into the potato mixture. Slice the steak against the grain and serve with the vegetables.