Make the French fried onions:
- Stir the soy milk and vinegar together in a medium bowl. Add the onion rings and submerge as many as possible. Allow onions to soak in the milk mixture for at least 30 minutes.
- Mix the flour, salt, pepper and cayenne together in medium bowl or large zipper bag.
- Remove the onions from the soy milk mixture, discarding any excess liquid. Transfer the onions to the flour mixture and toss to coat.
- Heat 1/4 cup of oil in a large skillet over medium-high heat. Once the oil begins to shimmer, add the onions in an even layer, only adding as many as you can without too much overlapping. Cook 2 minutes and gently toss once or twice with a spatula. Cook another 2 minutes and flip again, using the same technique. Once the onions begin to turn brown and crispy, after about another 2 minutes, remove and transfer to paper towel lined plate.
- If any onions remain after the first batch, repeat, adding a bit more oil to the skillet if needed.
Start the casserole:
- Add the olive oil to a medium skillet over medium heat. When it's hot, add the onion and sauté until softened, about 5 minutes. Add the garlic and the mushrooms. Sauté until the mushrooms are tender and release their juices, about 5 minutes more.
- Pour in the white wine and simmer, stirring occasionally, until liquid is reduced by half, about 2 minutes.
- Add the flour and stir to form a paste with the remaining liquid in the skillet. Slowly pour in the soy milk, stirring constantly. Stir in the soy sauce, pepper and green beans, then bring to a simmer and reduce heat to medium-low. Simmer, stirring occasionally, until the sauce is thick and the green beans are bright-green and tender-crisp, about 10 minutes.
- Preheat oven to 350°F while the sauce and green beans simmer.
- Transfer the bean mixture to a 9-inch baking dish and place it in the oven (or if you're using a cast-iron skillet, you can transfer it directly to the oven). Bake for 20 minutes and top with the French fried onions before serving.
The green bean casserole portion of this dish can be prepared in advance up to the baking step. Allow the casserole to cool after it's finished simmering and either transfer it to a storage container or refrigerate in the dish you plan to bake it in.
You can begin soaking your onions and mix the flour coating early in the day. Start the coating and frying process when your casserole has about 10 minutes of baking time left.