Bring a large pot of water to a boil. Once boiling, add the trimmed green beans and simmer for 2 1/2 to 3 minutes, or until tender-crisp. Transfer the blanched green beans to an ice bath to cool.
Meanwhile, preheat the oven to 375 degrees F. Prepare a 9”x13” baking dish.
Melt 1 tablespoon of butter in a large nonstick or stainless steel skillet over medium heat. Once melted, add the diced onions and mushrooms. Saute for 6-7 minutes, or until the mushrooms begin to brown and the onions are translucent and fragrant.
Add the minced garlic and stir in. Cook for 30-60 seconds, then transfer the mixture to a small bowl.
In the same pan, add the remaining butter. Melt over low heat. Once melted, dust the flour into the pan. Stir to combine, forming a thick, smooth paste (light roux). Cook 3-5 minutes, allowing the roux to bubble but not brown.
Carefully stir in the vegetable broth and bring to a low simmer. Stir, then add the vinegar. Continue stirring over medium-low heat until the sauce thickens, about 4-5 minutes.
Once thickened, add the Boursin cheese. Whisk or stir to incorporate. Add back the cooked onion and mushroom mixture, along with the blanched green beans. Fold together until well combined.
Transfer the green bean mixture to the 9"x13" pan. Spread in an even layer.
Cover and bake for 45 minutes. Remove from the oven and top with the fried onions. Return to the oven, uncovered, and cook under the broiler for 5 minutes before serving to brown the onions.