Preheat oven to 350F.
Heat a large skillet over medium-high heat with 1 turn of the pan of EVOO and 4 tablespoons of 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 about 1 minute then stir in the white wine and chicken stock, and reduce by half. Add the heavy cream and melt the Boursin cheese in the sauce, about 1 minute.
Add green beans to the skillet and combine. Pour into a medium-size casserole dish and top with onions.
Bake for 30 minutes, or until bubbly and brown.
For the Fried Onion Pieces:
Heat vegetable oil in a pot to 350F.
Soak onion slices in buttermilk and drain. Season flour with salt, pepper and paprika. Dredge onion slices in flour mixture. Fry in hot oil until golden brown, about 1-2 minutes.
Serve on top of green bean casserole.