Heat 2 tablespoons of olive oil in a large pot over medium-high heat. Once shimmering, add the ground venison in batches, breaking it apart as it cooks, and season generously with salt and pepper. Brown the meat for 8-10 minutes until deeply caramelized, then remove it to a plate. In the same pot, add the remaining 1 tablespoon of olive oil and sauté the diced onion for 3-4 minutes until softened and translucent. Add the minced garlic and cook for 30-60 seconds until fragrant. I like to use freshly minced garlic rather than pre-minced because it develops a cleaner, more vibrant flavor in the broth.
Return the browned venison to the pot, then add the chicken broth, diced tomatoes, crushed tomatoes, tomato paste, Italian seasoning, dried basil, and sugar. Stir well to combine, breaking up the tomato paste so it dissolves evenly into the broth. Bring the mixture to a boil over medium-high heat, then reduce the heat to medium-low, cover, and simmer for 20 minutes. This gentle simmering allows the flavors to meld and the venison to become tender while the tomato flavors deepen.
Break the lasagna noodles into 2-inch pieces and add them directly to the simmering soup. Stir occasionally to prevent sticking and ensure even cooking. Simmer uncovered for 10-12 minutes until the noodles are tender but still hold their shape. I find breaking the noodles rather than layering them creates a more appealing texture in the soup and ensures they cook evenly in the broth.
While the noodles finish cooking, combine the freshly shredded mozzarella, ricotta, and finely grated parmesan in a medium bowl, stirring gently until just combined. Once the noodles are tender, stir the butter into the soup pot to enrich the broth and add silky mouthfeel. Taste and adjust seasoning with salt and pepper as needed. The soup is now ready to serve.
Ladle the hot venison lasagna soup into bowls and top each serving with a generous dollop of the cheese mixture. The heat from the soup will soften the cheeses slightly, creating a creamy, luxurious texture throughout. Serve immediately while hot.