Bring 6 quarts water to a vigorous boil, add 1 tablespoon kosher salt, then cook the pasta to 1–2 minutes before package al dente; reserve 1 cup pasta water and drain.

Heat 1 tablespoon olive oil and 1 tablespoon butter in a 12-inch skillet over medium-high heat; add mushrooms in a single layer and brown undisturbed 3–4 minutes, then stir and cook until juices evaporate, about 4–5 more minutes; season lightly.

Add minced garlic to the mushrooms and cook 30–45 seconds until fragrant.

Lower heat to medium, add remaining 1 tablespoon olive oil and pour in 1/2 cup heavy cream; cook 1–2 minutes until it thickens slightly.

Turn off the heat and whisk in 1/2 cup reserved pasta water, then add 1 to 1 1/2 cups freshly grated Parmesan until melted, creating a glossy sauce; add more reserved water 1–2 tablespoons at a time to reach desired consistency.

Quickly wilt spinach in a separate skillet with 1 teaspoon olive oil over medium heat for 30–60 seconds until bright and just softened; drain any excess liquid, then fold spinach into the sauce.

Add drained pasta to the skillet and toss vigorously to coat, adding up to 1/4 cup more reserved pasta water if needed; finish with extra Parmesan and freshly ground black pepper, and serve immediately.