Cook the pasta in a large pot of salted boiling water according to package directions. Reserve 1 cup of pasta water, then drain.
While the pasta cooks, heat the olive oil and butter in a large skillet over medium-high heat. Add the mushrooms in a single layer and cook undisturbed for 3–4 minutes, then stir and cook 3–4 minutes more, until golden.
Add the garlic, Italian seasoning, and red pepper flakes and cook for 1 minute.
Add the spinach and stir until just wilted, about 1–2 minutes. Reduce the heat to low.
Add the drained pasta, then stir in the Greek yogurt, Parmesan, and ½ cup of pasta water. Toss until creamy, adding more pasta water a splash at a time if the sauce looks thick.
Season with salt and pepper and serve immediately with extra Parmesan.