Preheat the oven to 375 degrees F.
In a large saute or sauce pan, warm the oil over medium heat. Add the sausage and spread it around the bottom of the, allowing it to brown before turning with a spoon. If it’s easier, break up the sausage with a potato masher while it is in the pan to form bit-sized pieces. Continue to cook and stir until it is just cooked through, about 5-8 minutes.
Add all of the tomato sauce except a few tablespoons. (The reserved sauce will coat the bottom of the lasagna pan.) Stir to combine the sausage with the tomato sauce, then simmer for 10-15 minutes until the flavors meld. Taste, adding salt if necessary. (Many times the sausage has plenty of seasoning already.)
While the sauce simmers, make the cheese filling. In a medium mixing bowl, combine three cups of the mozzarella with the ricotta and 1/2 cup of the grated parmesan. Stir to combine. Taste, adding a pinch of salt and freshly ground pepper if needed.
Add the egg to the mixture and stir to incorporate.
Place the remaining cup of mozzarella and the remaining 1/4 cup of parmesan in a small bowl and toss to combine. Set aside.
To build the lasagna, spread the reserved tablespoons of tomato sauce across the bottom of a 8 X 12 or 9 X 13 baking dish. Place 4 of the lasagna noodles crosswise, allowing them to overlap a bit if necessary. (If the noodles you are using cover the length of the pan, place 3 of the noodles lengthwise to create the first layer.)
Drop ½ of the ricotta cheese mixture in spoonfuls across the noodles. Spread it out evenly with a spatula or even use your hands if necessary to create a fairly even layer. Pour 1/3 of the meat sauce over the cheese and spread with a spatula to cover.
Repeat with another layer of noodles, the remaining ricotta mixture, and ⅓ of the meat sauce.
Add the final layer of noodles, then the remaining sauce, ensuring the noodles are totally covered with the sauce.
Sprinkle the remaining mozzarella and parmesan over the entire lasagna.
Cover the pan tightly with a piece of aluminum foil. Place it on a baking sheet and bake for 45 minutes. Lift a corner of the foil and insert a knife in the lasagna. If it easily punctures the noodles, remove the foil and bake another 5 minutes until bubbly. If not, cover it again and cook for another 5-10 minutes.
Allow the lasagna to cool for 10-15 minutes before slicing. Serve immediately.