Preheat oven to 400°F.
Bring a large pot of water to a boil. Add pasta and cook according to package directions. Drain and set aside.
Meanwhile, heat 2 tablespoons oil in a large ovenproof skillet over medium heat. Add onion; cook, stirring, until softened, about 2 minutes. Add mushrooms and 1/4 teaspoon each salt and pepper; cook, stirring, until softened, 4 to 6 minutes. Add garlic; cook, stirring, for 1 minute. Transfer the mixture to a large bowl.
Add the remaining 1 tablespoon oil to the pan. Sprinkle in flour; cook, stirring, for 1 minute. Gradually add milk and mustard; cook, whisking constantly until the sauce thickens, 2 to 3 minutes. Remove from heat and stir in 1/2 cup Parmesan and the remaining 1/4 teaspoon each salt and pepper. Add the pasta, chicken, spinach and the mushroom mixture. Stir until well coated. Smooth the top and sprinkle with the remaining 1/4 cup Parmesan.
Bake until bubbling, about 15 minutes.