Pat the steak dry with a paper towel. Season the steak all over with 1/2 teaspoon salt, then let it rest at room temperature for 40 minutes or up to 2 hours.
Heat a medium stainless-steel or cast-iron skillet over high heat until very hot. Pat the steaks dry with paper towels again, then season the steaks with pepper.
Add 2 teaspoons of the olive oil to the pan, then add the steak and sear until is nicely browned on the bottom, checking after 1 minute and searing up to 3 minutes. Repeat on the other side, then continue cooking the steak, flipping with tongs every 1 minute, until the internal temperature has reached 110°F (43°C) for rare, 120ºF (49°C) for medium-rare or 130°F (54°C) for medium (steak will continue to cook for a bit once removed), 6 to 10 minutes depending on thickness. Remove the steak from the skillet, tent loosely with foil, and let it rest for 10 minutes.
Meanwhile, make the mushroom red-wine sauce: Using the same skillet set over medium-high heat, add 1 tablespoon of the butter, 1 heaping tablespoon minced shallot, and the mushrooms and cook, stirring occasionally, until the mushrooms have released their liquid and shallots are browned, 4 to 5 minutes. Add the red wine, then keep at a boil, stirring often until reduced and syrupy, 2 to 3 minutes. Add 1/4 cup of the broth and bring it to a boil, stirring often, until reduced by about one third. Remove the skillet from the heat and whisk in 1 tablespoon of the butter. Season the sauce with salt and pepper to taste.
Make the creamed spinach: In a clean skillet, melt the remaining 1 tablespoon butter and 2 teaspoons olive oil over medium heat. Add the remaining shallots and a pinch of salt. Cook until shallots are softened, about 3 minutes. Stir in the cream cheese, heavy cream, and remaining 1/4 cup broth; cook, stirring, until cream cheese is melted and smooth. Stir in the spinach; add the nutmeg and cayenne pepper and simmer over medium heat, stirring until spinach wilts and mixture thickens, 3 to 4 minutes. Remove from the heat and season with salt and pepper.
Slice the steak, divide it among 2 dinner plates, top the steak with the sauce, and serve with the creamed spinach on the side.