Heat oil in 12-inch nonstick skillet over medium heat until shimmering. Add mushrooms and salt, cover, and cook until mushrooms have released their liquid, about 5 minutes.

Uncover and continue to cook until mushrooms are well browned and crisp, 6 to 8 minutes. Using slotted spoon, transfer mushrooms to bowl. Wipe skillet clean with paper towels.

Add noodles to now-empty skillet and cook over medium heat until dry, about 3 minutes; transfer to bowl. Add ½ cup Pecorino, cream, cream cheese, and pepper to now-empty skillet and cook, stirring frequently, over medium-low heat until fully combined, 3 to 5 minutes. Gently fold in noodles and spinach and cook until heated through and spinach is wilted, about 2 minutes. Season with salt and pepper to taste and adjust consistency with hot water as needed. Sprinkle individual portions evenly with mushrooms and remaining ½ cup Pecorino before serving.