Preheat oven to 220°C (425°F). Season steak with salt and pepper. Heat a medium-size cast-iron skillet over medium-high heat. Add 1 tablespoon oil. Add steak, skin-on garlic, and rosemary sprig. Cook, turning once, until browned, 4 to 8 minutes.
Transfer skillet to oven, and roast to desired doneness, 3 to 6 minutes for medium. Transfer steaks to a cutting board, loosely tent with foil, and let rest at least 10 minutes before slicing.
Return skillet to medium-high heat (make sure to keep an oven mitt on the handle). Add mushrooms and remaining 2 tablespoons oil. Season with salt. Cook, stirring occasionally, until softened and starting to brown, 6 to 8 minutes. Add chopped garlic and chopped rosemary and cook, stirring, until fragrant, 1 minute.
Add mustard and wine. Cook, stirring, until thickened, 30 seconds. Transfer mushrooms to a plate. Add spinach to skillet and cook, tossing, until beginning to wilt, 1 to 2 minutes. Remove from heat, and stir in crème fraîche and mushrooms. Season with salt and pepper. Serve with sliced steak.