In a large skillet, heat the avocado oil over medium heat until shimmering. Add the green beans and sauté until bright green, about 4 minutes. Transfer to a bowl.
Add the butter to the skillet and let melt, then add the mushrooms and cook until browned and they have begun to release their juices, about 5 minutes. Add the garlic and cook 1 minute more.
Add the wine to the skillet and let cook until reduced by about two thirds. Stir in the heavy whipping cream and bring to a boil, then reduce the heat to low and simmer until thickened, 4 or 5 minutes.
Stir in the green beans and season to taste with salt and pepper. Transfer to a medium baking dish and sprinkle with the mozzarella. Bake 20 minutes. Remove and sprinkle with the crumbled parmesan crisps and bake another 5 minutes.