Cook linguine according to directions on the box adding 2 teaspoon of salt to the water. Make sure not to overcook the pasta. Strain pasta and set aside making sure to reserve ½ cup of the pasta water.
In a large stovetop skillet or pot (I will link the one I have below) Melt the butter.
Clean and cut off broccoli florets into bite-size and put them into the pot. Season them with salt, pepper, and paprika. Sautee on medium heat for about 8-10 minutes or until tender.
Grate in the garlic and stir for another minute or two.
Stir in the heavy cream, broth, ½ cup of mozzarella cheese, and ¼ cup of parmesan . Turn heat down to low and let lightly simmer for about 8-10 minutes stirring often.
When the pasta is finished add ½ cup of pasta water and stir well. Taste and add more salt and pepper if needed.
Toss the cooked linguine into the sauce and top with remaining cheeses. Put the pasta under the broiler in your oven for about 5 minutes to melt the cheeses. You can also do this on the stovetop by adding a lid and cooking on low heat until the cheese is melted.