Bring a pot of salted water to a boil. Heat oil in a large skillet over medium-high, crumble in sausage, and cook until browned, 6-8 minutes. Meanwhile, boil pasta until al dente, reserving 1 cup pasta water.

Reduce heat to medium-low, add garlic and red pepper flakes, and cook 30 seconds.

Pour in broth, scrape up browned bits, then add cream and simmer until thickened, 3-4 minutes. Stir in Parmesan, salt, and pepper.

Return sausage to the skillet, add spinach by the handful, and stir until wilted, about 1 minute.

Toss in drained pasta, adding pasta water as needed until sauce coats every piece. Serve immediately with extra Parmesan.