When the mushrooms have released their liquid, stir in the garlic and cook for 30 seconds. Add the Marsala and cook over medium-high heat, stirring frequently, until almost all of it evaporates, about 4 minutes.
Meanwhile, pour about 1/4 of the jar of Alfredo sauce into the bottom of the first baking pan, then another 1/4 of the jar into the second pan. Rotate the pans so that the sauce coats the pan bottoms. Place two noodles side-by-side over the sauce in each pan.
When the Marsala has evaporated, remove the skillet from the heat. Spoon about 1/4 of the mushroom-onion mixture over the noodles in the first pan. Repeat with the second pan. Sprinkle about 1/2 cup of the mozzarella over the mushrooms in each pan.
Place two more noodles side by side over the cheese in each pan. Spoon half of the cottage-cheese mixture over the noodles in each pan, spreading to the edges of each pan. Spoon half of the diced tomatoes and their juices evenly over the cottage cheese. Top this layer in each pan with two more noodles. Divide the remaining mushroom mixture on top of the noodles in each pan. Divide the remaining Alfredo sauce between the pans, spreading sauce to cover the noodles completely. Sprinkle the remaining mozzarella and the Parmesan evenly over both pans.
Cover with nonstick aluminum foil and bake 50 minutes. Remove from the oven, uncover and, if eating now, bake 5 more minutes. Remove from the oven and let the lasagna rest for 10 to 15 minutes before serving.
If freezing, uncover the pan, cool to room temperature, recover with foil and freeze for up to 1 month. Or refrigerate for up to 48 hours. Defrost for 24 hours in the refrigerator, and allow the lasagna to stand at room temperature, uncovered, for 30 minutes. Reheat, covered with the foil, at 325 degrees for 30 minutes. Uncover, raise the heat to 375 degrees, cook for 5 minutes more, and serve.