Combine broth and bay leaves in a small saucepan; bring to a boil over high heat. Once boiling, add cashews and then remove pan from heat. Let the cashews soak in the hot broth for at least 30 minutes.
Meanwhile, preheat oven to 425ºF. Place torn sourdough bread in a large bowl and drizzle with 2 Tbsp. olive oil and a generous pinch of salt. Toss well, using your hands to rub the oil into the crevices of the bread. Set aside.
Heat remaining 2 Tbsp. olive oil in a large skillet over medium-high. Once hot, add mushrooms and onion; cook 8 to 10 minutes, stirring occasionally, until softened. Stir in garlic, thyme, and nutmeg; cook 1 more minute, until aromatic. Stir in vinegar and season with 3/4 tsp. kosher salt and 1/2 tsp. black pepper; cook 1 to 2 more minutes.
Remove (and discard) bay leaves from cashew-broth mixture and carefully transfer mixture to a high-power blender. Add remaining 1/2 tsp. salt and blend for about 60 seconds, until the consistency is smooth and creamy (it should resemble milk). Carefully pour mixture into pan with mushrooms and stir to combine. Bring to a simmer.
Grease a 3-quart baking dish with non-stick cooking spray. Add green beans to pan, and carefully pour creamy cashew mushroom mixture overtop. Use a set of tongs to toss.
Scatter sourdough bread overtop and transfer pan to the oven. Bake, uncovered, for 20 to 25 minutes, or until the sourdough is golden and casserole is bubbly. Let stand 5 minutes before serving.