Preheat the oven to 350 degrees F. In a medium pot heat 1 tablespoon of the oil over medium-high heat. Add the garlic and cook until fragrant, about 1 minute. Add the tomatoes with their juices, oregano, 3/4 teaspoon salt and 1/4 teaspoon pepper. Bring to a simmer and continue to cook until the tomatoes soften and the sauce thickens, 20 minutes, breaking up the tomatoes with a spoon as they soften. Reserve 2/3 cup of the sauce for assembling the lasagna and set the rest aside.
Meanwhile, heat the remaining 1 tablespoon oil in a large skillet over medium-high heat and add the mushrooms. Cook until the mushrooms have released their liquid, 5 to 7 minutes. Add the sausage and continue to cook, breaking the sausage up with a spoon, until the sausage is browned and cooked through, another 10 minutes. Add the tomato sauce and stir to combine.
Thinly slice about a quarter of the mozzarella and set aside; grate the rest on a box grater. Combine the grated cheese with the ricotta and the egg. Spread the reserved 2/3 cup tomato sauce in the bottom of a 9-by-13-inch baking dish. Top with 1 pasta sheet, a third of the ricotta mixture and a third of the meat sauce. Repeat with the remaining pasta, ricotta and meat sauce. Cover with foil and bake until the sauce is bubbling around the edges and the pasta is tender, about 45 minutes. Remove the foil, top with the sliced mozzarella and continue to bake until the cheese is melted, 15 minutes more. Let stand for 20 minutes and sprinkle with chopped parsley before serving with a green salad and some garlic bread.
You can assemble the lasagna up to 1 day before serving and refrigerate. You may need to add a few minutes to the baking time. Make it a Meal: Crudite with Infused Olive Oil and Balsamic, Truffled Pots de Creme