Step 1Bring a large pot of water to a boil. Add the green beans and 1 tablespoon salt. Boil, stirring occasionally, until brighter green and tender with a little bite left, about 5 minutes. Drain, rinse under cold water until room temperature, and drain again.
Step 2Meanwhile, heat 1/4 cup oil in a large Dutch oven over medium-high heat. Add the onion and a big pinch of salt and cook, stirring often, until almost translucent, 5 to 10 minutes.
Step 3Add the mushrooms, the remaining 2 tablespoons oil and a pinch each of salt and pepper. Cook, stirring often, until the mushrooms soften, about 5 minutes. Add the flour, reduce the heat to medium, and cook, stirring, until the raw flour smell dissipates, 2 to 3 minutes. Add the oat milk, 1/4 cup at a time, stirring well to incorporate each addition before adding the next. You want the sauce to stay smooth and not get lumpy. Bring the sauce to a gentle simmer and cook until creamy, about 3 minutes. Season to taste with salt and pepper.
Step 4Return the green beans to the pot in which they cooked and add the mushroom sauce. Fold until evenly coated, then transfer to a serving platter. Sprinkle the fried onions all over the top right before serving.