In a large, deep skillet, heat 4 tablespoons EVOO, 4 turns of the pan, and half the butter over medium heat. Add the bread and garlic and lightly toast the bread, 7 to 8 minutes. Transfer the bread and garlic to a 9-by-13-inch casserole. Add the remaining 3 tablespoons EVOO, 3 turns of the pan, to the skillet; stir in the mushrooms, parsley and thyme and cook until tender, 7 to 8 minutes. Season with salt and pepper.
Melt the remaining 2 tablespoons butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute. Add the milk and cook, whisking, until thickened, about 5 minutes. Stir in the Parmigiano-Reggiano and season with salt and pepper and the nutmeg. Keep warm.
Preheat the broiler to high. Stir the wine into the mushrooms for 1 minute. Add the spinach and wilt. Spoon the vegetable mixture into the casserole. Pour the white sauce on top; cover with the provolone. Broil until bubbly, 2 to 3 minutes.