Heat the olive oil and butter in a large Dutch oven or stockpot over medium heat. Once the butter has melted, add the diced onion and cook for 3–4 minutes, stirring occasionally, until softened. Add the minced garlic and cook for another 30 seconds until fragrant.
Sprinkle the flour over the onion and garlic mixture. Whisk continuously for 1–2 minutes to cook out the raw flour taste and create a smooth base for the soup.
Slowly pour in the chicken broth, whisking as you go to prevent lumps from forming. Stir in the heavy cream, whole milk, chicken bouillon powder if using, onion powder, garlic powder, paprika, salt, and black pepper. Bring the mixture to a gentle simmer.
Add the chopped broccoli, shredded carrots, and broken lasagna noodles to the pot. Stir well to distribute the ingredients evenly and make sure the noodles are submerged in the liquid.
Reduce the heat to medium-low, cover the pot partially, and simmer for 15–20 minutes, stirring occasionally to prevent the noodles from sticking to the bottom. Cook until the lasagna noodles are tender and the broccoli and carrots are cooked through.
Turn off the heat and gradually stir in the freshly grated cheddar cheese, a handful at a time. Allow each addition to melt before adding more. Stir gently until the soup becomes creamy and the cheese is fully incorporated.
Taste the soup and adjust the salt and pepper as needed. If the soup is thicker than you prefer, add a little extra broth or milk until it reaches your desired consistency.
Ladle the broccoli cheddar lasagna soup into bowls and serve hot. Add a little extra shredded cheddar or freshly cracked black pepper on top, if desired.