Soak bread slices in warm water in a bowl for 5 minutes; squeeze bread dry and transfer to a large bowl.
Melt butter in a skillet over medium heat; cook and stir onion and garlic in hot butter until onion is translucent, about 5 minutes. Add onion and garlic to bread. Stir 1 1/2 cup mozzarella cheese, eggs, Parmesan cheese, and Italian seasoning into bread and onions until thoroughly combined. Crumble ground beef into mixture; mix until thoroughly combined. Season with salt and black pepper.
Form ground beef mixture into 1-inch meatballs. Heat vegetable oil in a large skillet over medium heat. Pan-fry meatballs until browned, working in batches if necessary, about 10 minutes, turning often. Set meatballs aside.
Bring a large pot of lightly salted water to a boil. Cook lasagna in the boiling water, stirring occasionally until cooked through but firm to the bite, about 8 minutes. Drain.
Preheat oven to 350 degrees F (175 degrees C).
Combine prepared spaghetti sauce, tomato paste, and Cheddar cheese soup in a bowl. Combine ricotta cheese, cottage cheese, 2 cups mozzarella cheese, and parsley in a separate bowl.
Spread 1 cup of sauce mixture into the bottom of a 9x13-inch baking dish. Layer 1/3 the lasagna noodles over the sauce, overlapping if necessary. Arrange meat balls over noodles. Place 1/3 the lasagna noodles over meat balls, and spread with cheese mixture in an even layer. Top cheese layer with remaining 1/3 the lasagna noodles and spread remaining sauce over top layer of noodles. Sprinkle top with 1 cup mozzarella cheese.
Bake lasagna in the preheated oven until casserole is bubbling and the topping is melted, about 1 hour.