Bring a large pot of salted water to a rolling boil. Add the pappardelle or fettuccine pasta and cook according to package directions until al dente. Drain, reserving about 1/2 cup of the pasta cooking water.

While the pasta cooks, heat 2 tablespoons of olive oil in a large skillet or Dutch oven over medium-high heat. Add the sliced mushrooms in a single layer (you may need to do this in batches to avoid overcrowding). Cook without stirring for 3-5 minutes until deeply browned on one side, then stir and continue cooking until all liquid has evaporated and mushrooms are tender and caramelized, about 8-10 minutes total. Remove the cooked mushrooms from the skillet and set aside.

Reduce the heat to medium. Add the 3 tablespoons of vegan butter to the same skillet. Once melted, add the finely diced yellow onion and cook, stirring occasionally, until softened and translucent, about 5-7 minutes. Add the minced garlic and cook for another minute until fragrant.

Sprinkle the 3 tablespoons of all-purpose flour over the onion and garlic. Stir well to coat, forming a roux. Cook for 1-2 minutes, stirring constantly, to cook out the raw flour taste.

(Optional) Pour in the 1/2 cup of dry white wine, scraping up any browned bits from the bottom of the pan with a wooden spoon. Let it simmer for 1-2 minutes until slightly reduced.

Gradually whisk in the 2 cups of vegetable broth, a little at a time, ensuring no lumps form. Bring the sauce to a gentle simmer and cook for 3-5 minutes, whisking occasionally, until it has thickened to your desired consistency.

Stir in the 1 tablespoon of Dijon mustard and 1 tablespoon of Tamari (or soy sauce).

Reduce the heat to low. Stir in the 1 cup of vegan sour cream until fully combined and warmed through. Do not boil the sauce after adding the sour cream, as it may curdle.

Return the cooked mushrooms to the skillet with the sauce. Stir in the 1/4 cup of chopped fresh dill or parsley. Season with salt and freshly ground black pepper to taste. If the sauce is too thick, add a splash of the reserved pasta water until it reaches your preferred consistency.

Add the drained pasta directly to the skillet with the sauce and toss to coat, or serve the stroganoff sauce over individual portions of pasta. Garnish with additional fresh dill or parsley, if desired. Serve immediately.