Heat the olive oil and butter in a large Dutch oven or stockpot over medium heat. Once the butter melts, add the diced onion and cook for 3–4 minutes, until softened. Add the garlic and cook for another minute.
Sprinkle the flour over the onion mixture and whisk until well combined. Cook for 2–3 minutes, stirring frequently.
Slowly pour in the chicken or vegetable broth, heavy cream, and whole milk while whisking continuously to create a smooth, creamy base.
Stir in the chicken bouillon, onion powder, garlic powder, paprika, salt, and black pepper.
Add the broccoli, shredded carrots, and broken lasagna noodles. Increase the heat to medium-high and bring the soup to a gentle boil.
Reduce the heat to medium-low, cover, and simmer for about 15 minutes, stirring occasionally so the noodles do not stick. Cook until the noodles are al dente and the vegetables are tender.
Turn off the heat and gradually stir in the freshly grated cheddar until completely melted and the soup is smooth and creamy.
Taste and adjust the seasoning with additional salt and pepper if needed. Ladle into bowls and serve warm.