Rinse mushrooms; trim off and discard discolored stem ends. Thinly slice mushrooms. In a 5- to 6-quart pan, combine mushrooms, onion, and olive oil. Cover and stir often over medium-high heat until mushrooms are juicy, about 4 minutes. Uncover, add parsley, and stir often over high heat until vegetables are lightly browned, 5 to 8 minutes. Pour mixture into a bowl.
Add beef and sausages to pan. Over high heat, crumble meats with a spoon and stir often until well browned, 10 to 15 minutes; discard fat. Add mushroom mixture, thyme, vinegar, tomato paste, tomato sauce, and broth; stir until boiling. Reduce heat and simmer, stirring often, until most of the liquid has evaporated, about 30 minutes.
Spoon 1 cup béchamel sauce into a shallow 9- by 13-inch casserole (3 qt.) and spread evenly over bottom. Arrange 1/4 of the pasta evenly over béchamel sauce to cover as much of the casserole bottom as possible.
Spoon 1/3 of the meat sauce over pasta in casserole and spread level. Drizzle meat sauce with about 1/2 cup béchamel and sprinkle with about 1/4 cup cheese. Cover cheese with another 1/4 of the pasta and top with another 1/3 of the meat sauce, 1/2 cup béchamel, and 1/4 cup cheese. Arrange a third layer of pasta over cheese and cover with remaining meat sauce, 1/2 cup béchamel, and 1/4 cup cheese. Cover with the last 1/4 of the pasta and spread remaining béchamel over pasta to moisten evenly; sprinkle with remaining cheese and the nutmeg. Cover pan tightly with foil.
Bake in a 350° regular or convection oven until sauce is bubbling vigorously and pasta edges are beginning to brown (lift a corner of foil to check), about 40 minutes.
Let stand about 10 minutes; cut into rectangles and serve with a wide spatula.