Preheat the oven to 350 degrees F.
Cook the green beans in heavily salted boiling water until just tender crisp, 2 to 3 minutes. Use a slotted spoon to transfer the beans to a salted ice bath. Once cooled, drain well and transfer to a large bowl.
Toss the shiitakes with the olive oil, sprinkle with salt and pepper and spread on a baking sheet. Roast until golden brown, 8 to 10 minutes. Transfer to the bowl with the green beans.
In a large 12-inch cast-iron skillet, melt the butter over medium heat. Add the onion and saute until soft, 6 to 8 minutes. Add the flour and cook until golden brown, 2 minutes more. Pour in the mushroom broth, and bring up to a simmer. Cook until the sauce thickens, 6 to 8 minutes. Stir in the cream cheese, heavy cream, thyme, salt and pepper. When the cream cheese is melted and the sauce is smooth, remove from the heat and pour into the bowl with the green beans and shiitakes. Set aside.
Spread the Goat Cheese Smashed Potatoes in the cast-iron skillet. Pour the green bean mixture over the mashed potatoes and top with the fried onions. Bake until all the ingredients are hot and bubbly, 20 minutes.
Place the potatoes and yams in a large saucepan, cover with cold water and season with salt. Bring to a boil and cook until fork-tender, 15 to 20 minutes.
Drain the potatoes and yams and place in a stand mixer fitter with a whisk attachment. Add the goat cheese, butter and garlic and mix on low just until smooth, being careful not to over-mix or the potatoes will become gummy. Season with salt and pepper. If necessary, add the milk to loosen.