Place a large Dutch oven or heavy-bottomed pot over medium heat. Melt the butter and add the diced onion. Sauté for 4 to 5 minutes until the onion becomes translucent and soft. Stir in the minced garlic and cook for another 60 seconds until fragrant, being careful not to let the garlic brown or burn.

Sprinkle the all-purpose flour over the sautéed onions and garlic. Whisk constantly for about 2 minutes to cook out the raw flour taste. Slowly pour in the broth, whisking vigorously to ensure no clumps form. Once the broth is incorporated, stir in the heavy cream, smoked paprika, and nutmeg. Bring the mixture to a gentle simmer.

Add the broken lasagna noodle pieces into the simmering liquid. Stir them well to prevent the pieces from sticking together. Reduce the heat to medium-low and let the noodles simmer for about 8 to 10 minutes. Stir occasionally to ensure the pasta cooks evenly and doesn’t stick to the bottom of the pot.

When the noodles are nearly al dente (slightly firm), fold in the chopped broccoli florets. Continue to simmer for another 3 to 5 minutes until the broccoli is tender-crisp and bright green. The residual heat from the soup will cook the broccoli perfectly without making it mushy.

Remove the pot from the heat. Gradually stir in the shredded cheddar cheese, one handful at a time, until the sauce is completely smooth and cheesy. Season with salt and black pepper to taste. If the soup is too thick, add a splash of broth to reach your desired consistency. Ladle into bowls and serve immediately.