Bring a large pot of water to a boil and cook the pasta according to package directions.
In a large measuring cup or medium bowl whisk together the vegetable broth, non-dairy milk, flour, dried thyme, salt, and pepper. Set aside.
Melt the vegan butter in a large skillet or pot. When hot add the onion and garlic and sauté for about 5 minutes until the onion turns translucent and begins to brown. Reduce the heat to medium and add the mushrooms. Continue to cook for about 5 minutes until the mushrooms have softened and begin to release their juices.
Pour in the broth and non-dairy milk mixture and continue to cook, stirring as needed for about 3 to 5 minutes until the sauce thickens. If the sauce gets too thick, add more vegetable broth or water if needed to thin. If the sauce is too thin, simply cook a little longer until you reach desired consistency.
Add in the cooked pasta and toss well to combine. Divide among plates and serve, garnishing with parsley and Parmegan to taste.
Traditionally stroganoff is served over egg noodles which aren't vegan, so I just subbed for an egg-free pasta noodle. Today I used tagliatelle. You could alternatively serve this over rice.