Bring 4 quarts salted water to a boil. Add lasagne noodles; boil 10 minutes, until tender but still firm. Plunge noodles into a bowl of cold water to stop cooking. Drain, and lay side by side on a towel to dry.
Next, make filling. In a bowl, combine beaten egg with ricotta and mozzarella cheese; set aside. In a large skillet, heat oil, and saute onions and garlic until tender (about 5 minutes). Add ground beef and break up with a spoon. Cook until browned; drain off excess fat. Stir in 1 cup of the parsley, basil, salt, oregano, and pepper. Add egg-cheese mixture, and stir to blend.
Pour 1 jar of the tomato sauce into the bottom of a 3-quart oblong lasagne pan (or use two 7-½ by 11-½ inch baking dishes). Next, spread each lasagna strip with a little less than ¼ cup of filling, then roll up, jelly-roll fashion. Cut each roll in half and place each half in the baking dish with the cut ends down.
Prepare bechamel (white) sauce. Melt butter in saucepan over medium heat. Stir in flour, blending until smooth. Gradually stir in milk. Cook, stirring constantly, until mixture boils and thickens. Stir in 1 cup of parmesan cheese, salt, and pepper. Spoon bechamel sauce over lasagna rolls.
Pour remaining tomato sauce over bechamel sauce. Sprinkle top with remaining chopped parsley and Parmesan cheese. Cover with foil, bake at 400 degrees for 20 minutes. Remove foil: bake 10 minutes more, until bubbly. Cool 10 minutes.