Cook the pasta sheets, in batches, in a large saucepan of salted boiling water according to the packet instructions until they are just al dente. Drain, refresh in cold water and lay on a tea towel to drain thoroughly. Place 4 sheets in the base of a well-oiled, large ovenproof dish.
Meanwhile, melt the butter with the oil in a saucepan over a medium heat. Add the onions and cook, stirring, for 3 minutes. Add the garlic and cook, stirring, for 1 minute. Add the mushrooms, increase the heat to high and cook, stirring frequently, for 5 minutes. Add the cream, wine and thyme, and season with salt and pepper. Simmer for 4 minutes.
Place a generous spoonful of the mushroom mixture on each pasta sheet and add some red pepper slices and half the spinach. Top with the remaining pasta sheets. Add the remaining spinach, a slice of mozzarella and top with a little more mushroom mixture. Finish with some Parmesan shavings.
Place under a preheated very hot grill and cook for 5 minutes, or until the mushroom mixture is bubbling and the Parmesan is golden. Serve immediately.