Prepare topping: Stir together ricotta, Parmesan, nutmeg, and ¼ teaspoon of the salt in a medium bowl. Cover and refrigerate until ready to serve.
Brown ground beef: Heat oil in a large Dutch oven over medium-high. Add ground chuck. Cook, stirring and breaking up into pieces, until browned and just cooked through, about 3 minutes. Using a slotted spoon, transfer beef to a small bowl lined with paper towels; set aside.
Cook onions and garlic: Do not wipe Dutch oven clean. Add onion to Dutch oven. Cook over medium-high, stirring occasionally, until softened and starting to caramelize, about 4 minutes. Add garlic. Cook, stirring constantly, until fragrant, about 1 minute. Stir in diced tomatoes and stock; bring to a boil over high. Stir in sugar and remaining 1 teaspoon salt.
Add pasta: Add lasagna pieces. Cook, stirring occasionally, until noodles are just cooked through and almost al dente, about 8 minutes. Stir in browned beef. Remove from heat, and stir in vinegar.
Garnish and serve: Divide soup evenly among 4 bowls; top with mozzarella and dollops of reserved ricotta mixture. Top bowls with basil leaves before serving.
