Melt the first 2 tablespoons of butter in your pot over medium heat. Add the diced onion and cook until it turns soft and translucent, about four minutes. Stir in the minced garlic and cook for another minute, just until it smells fragrant.
Toss the shredded or chopped carrots into the pot with the softened onion and garlic. Let them cook for about three minutes so they start to soften slightly. This gives the carrots a head start before the liquid goes in.
Push the vegetables to one side and melt the remaining 3 tablespoons of butter in the empty space. Sprinkle in the flour and whisk constantly for about two minutes. This step is what thickens the soup without any heavy cream at all.
Slowly pour in the warm broth while whisking so the roux doesn't clump. Once it's smooth, stir in the milk. Bring the mixture to a gentle simmer, stirring occasionally so nothing sticks to the bottom.
Add the smoked paprika, onion powder, garlic powder, white pepper, and a pinch of kosher salt. Stir everything together so the seasoning spreads evenly through the broth. You can always adjust the salt again later.
Drop the broken lasagna noodles into the simmering soup, followed by the broccoli florets. Cover the pot and let everything cook for about fifteen minutes, stirring once or twice so the noodles don't stick together.
After fifteen minutes, taste a piece of noodle to make sure it's tender but not falling apart. The broccoli should be bright green and just fork-tender at this point. If the noodles need more time, cover and simmer for a few more minutes.
Turn off the heat completely before adding the cheese, since direct high heat can make it separate instead of melting smoothly. Stir in the shredded cheddar and mozzarella a handful at a time until fully melted and the soup looks glossy.
Give the soup a final taste and add more salt or pepper if needed. If it feels too thick, stir in a splash of more milk. If it feels thin, let it sit uncovered for a few minutes to reduce slightly.
Ladle the soup into bowls right away while it's hot and the cheese is still stretchy. This soup thickens as it sits, so serving it soon after cooking gives you the best texture.