Whisk broth, Parmesan, cream, nutmeg, and 1/2 teaspoon pepper together in a bowl; set aside. Melt 2 tablespoons butter in a 12-inch nonstick skillet over medium-high heat. Add gnocchi and cook until lightly browned, about 5 minutes; transfer to a plate.
Melt remaining 1 tablespoon butter in now-empty skillet over medium-high heat. Add mushrooms and 1/4 teaspoon salt and cook until golden brown, about 10 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Add broth mixture and gnocchi and bring it to simmer. Cook, stirring occasionally, until the sauce has thickened, 5 to 7 minutes. Sprinkle with basil and serve.
Notes
The partially cooked, vacuum-packed gnocchi found in the pasta aisle work best here, but refrigerated or frozen gnocchi can also be used.