Cut off the stem ends of the green beans and cut the beans in half. Cut any large cauliflower florets into smaller pieces so that all the cauliflower pieces are a similar size.
When the water boils, boil the green beans and cauliflower for 5 minutes.
Place the green beans and cauliflower in a colander, rinse with cold water to stop the cooking, and drain well. Set aside.
Preheat the oven to 375 F.
Heat the olive oil on medium heat in a large skillet.
Add the shallot, garlic, and mushrooms and cook for about 5 minutes, until the onions are translucent and the mushrooms have reduced in size.
Add the flour and cook for 2 minutes while stirring to evenly coat the mushrooms.
Pour in the wine. While it boils gently, stir for about 1 minute, until the liquid is almost gone.
Pour in the broth and tahini and carefully stir to dissolve all of the tahini.
Add the salt and pepper and let the mixture simmer for about 5 minutes, stirring occasionally, until it is slightly thickened.
Add the green beans and cauliflower to the skillet and stir gently to coat them with the sauce.
Pour the mixture into an oiled baking dish.
Top the dish with the fried onions.
Bake for 15 minutes, until the sauce is hot and bubbly and the onions are lightly browned.
Let the casserole cool for about 5 minutes before serving.