Pre-heat the oven to 350°F.
Heat a large skillet over medium-high heat with one turn of the pan of EVOO and 4 tablespoons butter. Once hot, add the mushrooms and cook, stirring every now and then until brown, about 5 minutes. Add the shallots, garlic, thyme, salt and pepper. Continue to cook until the shallots start to get tender, about 4-5 minutes.
While the mushrooms and shallots cook, place the green beans in another skillet. Add an inch or two of water, salt it and bring to a boil. Cook the beans for 2-3 minutes, drain and shock in ice water.
When all the vegetables are tender, sprinkle the flour over the mushrooms and shallots. Cook for about 1 minute, then stir in the white wine and chicken stock; reduce by half. Add the heavy cream and melt the Boursin cheese into the sauce, about 1 minute.
Add the green beans to the skillet and combine. Pour into a medium size casserole dish and top with the onions.
Bake for 30 minutes, or until bubbly and brown.
For the fried onion pieces:
In a pot, heat vegetable oil 350°F.
Soak the onion slices in buttermilk and drain. Season the flour with salt, pepper and paprika. Dredge the onion slices in flour mixture. Fry in the hot oil until golden brown, about 1-2 minutes.
Serve on top of the green bean casserole.