Fill a pasta pot with pasta and steamer inserts three-quarters of the way full with salted water. Place over medium-high heat and bring to a rapid boil. Cook the pasta in the pasta insert according to the package instructions.
Meanwhile, fill the steamer insert with the asparagus, snap peas and green beans. When there is 4 minutes left on the pasta, add the steamer insert to the top of the pot and cover with the lid.
Carefully remove the steamer insert and set aside. Remove the pasta insert, shaking the excess water back into the pot, and set aside. Reserve 1 cup pasta water and dispose of the remaining water.
Return the pasta pot to high heat and add the olive oil, butter, chili flakes, tomatoes, garlic and some salt and pepper and cook, stirring occasionally, until the garlic is fragrant and beginning to brown, about 1 minute.
Deglaze the pot with the white wine and 1/2 cup reserved pasta water. Stir in the drained pasta, steamed vegetables and spinach. Cook until the spinach has wilted and the sauce has slightly thickened, adding pasta water as needed.
Stir in three-quarters of the Parmesan and season to taste. Serve, garnished with the remaining Parmesan and torn basil.