Heat the oil in a large high-sided skillet or pot over medium heat. Add the onion and mushrooms and saute until browned and the liquid is mostly released, about 8 minutes. Add the garlic and hot pepper flakes and cook for 1 minute, until fragrant. Stir in 1 teaspoon salt.
Stir in the vegetable broth, pasta, and remaining 1 teaspoon salt. Bring to a simmer, then reduce heat to medium low. Cover the pot and cook for 11 minutes, until the pasta is al dente, stirring halfway through. Turn off the heat.
Remove the lid and add the spinach. Cover and let sit for 2 minutes. Stir in the nutritional yeast and lemon zest and season to taste with salt and/or hot pepper flakes. Serve with vegan cheese.