Preheat the oven to 375° F, boil a large pot of salted water, thinly slice the onions, and slice the mushrooms. In a measuring cup or mixing bowl, combine the chicken broth with the oat milk and set aside.
Heat olive oil in a large pan over medium heat. Add the onions, stirring occasionally for 10-12 minutes, until they start to turn golden brown. Transfer to a medium sized mixing bowl and toss with 2 tbsp melted butter, 1/4 cup flour and panko until fully coated. Spread the onions onto a sheet pan and bake for 5-10 minutes.
Meanwhile, cook the green beans In the large pot of salted water for 4-5 minutes until almost tender, then drain and transfer to a large bowl filled with ice water to stop them from cooking more.
Meanwhile, heat 2 tbsp butter in the same pan you used for the onions, over medium heat. When hot, add the mushrooms and sauté for 3-4 minutes, until they start to release their juices. Then add the garlic, 1 tbsp thyme, salt, and pepper. Continue cooking for 5-8 minutes, or until the mushrooms have softened and started to brown. Transfer to a bowl or plate and set aside.
Using the same pan, melt 3 tbsp of butter and quickly whisk in 1/4 cup of flour. Continue whisking until a thick paste forms, then reduce the heat to medium-low and slowly pour in the broth combined with the oat milk. Whisk quickly! Whisk in 1/2 tbsp fresh thyme, garlic powder and salt and pepper to taste. Simmer for 5 minutes. Remove sauce from heat and whisk in parmesan cheese until fully combined.If you notice the sauce is way too thick, slowly add chicken broth - it should be smooth and creamy but still thick, not thick and jelly.
Grease a 9X13 baking dish with olive oil. Add the green beans, mushrooms, and gravy. Gently mix to combine and cover with foil. Bake 20-25 minutes, remove the foil, place the onions on top and bake 5-10 minutes. For extra crispy onions, place under the broiler for 1-2 minutes.