Drain the cashews, then place them in a blender with 3/4 cup of the vegetable broth. Puree until smooth, thick, and creamy. Depending upon your blender, this may take several minutes and you may need to stop and scrape down the blender a few times. The mixture will be the consistency of a thin paste. Set aside.
In a large, deep skillet or a Dutch oven with a tight-fitting lid, heat the olive oil over medium. Add the onion and cook until beginning to soften, about 5 minutes.
Add the mushrooms. Continue to cook, stirring occasionally, until the mushrooms are browned and softened and have given up their liquid, about 10 to 12 minutes.
Stir in the thyme, smoked paprika, black pepper, salt, and garlic. Cook, stirring to coat the mushrooms, until the garlic is fragrant, about 30 seconds.
Stir in the soy sauce. Let cook until the soy sauce evaporates, about 30 seconds to 1 minute.
Sprinkle the flour over the top, then stir so that it coats the mushrooms. Continue stirring until all of the white bits of flour disappear.
Pour in 1 cup of the remaining broth. Stir and work any stuck-on bits off the bottom of the pan.
Add 1 1/2 cups of the remaining broth and the noodles. Stir to combine and cover the pan. Bring the liquid to a boil, then immediately remove the lid and stir again. Re-cover, then reduce the heat to a low simmer. Let simmer until the noodles are al dente and most of the liquid is absorbed, about 9 to 10 minutes. Check every minute or so to give the noodles another stir and run a rubber spatula or wooden spoon along the bottom of the pan to prevent sticking, recovering the pan each time. If at any point, the noodles look dry, splash in more of the remaining broth. Adjust the heat as needed so that you maintain a gentle simmer.
Once the noodles are tender, remove the pan from the heat. Stir in the blended cashews, nutritional yeast, and vinegar. If the sauce is too thick at any point, splash in more broth to loosen it (I ended up adding almost all 4 cups by the end.) Taste and adjust seasonings as desired. Sprinkle with fresh parsley and serve hot.
*In lieu of the blended cashews, you can use 2/3 cup of a thick, non-dairy yogurt (bring it to room temperature and let the noodles cool for a few minutes before stirring it in) or vegan cream cheese, cut into pieces and softened to room temperature; if you go this route, add 1 tablespoon of fresh lemon juice or an additional 1/2 to 1 tablespoon vinegar.
**Egg noodles are more traditional for stroganoff, but I find it harder to locate the whole wheat variety, compared to easy-to-find rotini, and rotini holds up a bit better when cooking a one-pan pasta too. If you'd like to use egg noodles, start with only 2 cups additional broth in Step 6; the pasta will cook more quickly, in about 7 minutes. Keep an eye on it and add more liquid as needed.
TO STORE: Refrigerate stroganoff in an airtight storage container for up to 4 days.
TO REHEAT: Gently rewarm leftovers in a Dutch oven on the stovetop over medium-low heat.
TO FREEZE: Freeze leftovers in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.