Boil the pasta and chopped broccoli together in a large pot of salted water until tender. Drain, reserving about a half cup of the starchy pasta water.
In a separate saucepan, melt the butter over medium heat and sauté the finely chopped shallot until soft.
Sprinkle the flour over the butter and shallots, stirring continuously to form a roux.
Slowly pour in the milk and whisk continuously until the sauce begins to thicken. Stir in a splash of the reserved starchy pasta water to help thin it out slightly.
Add the shredded cheddar cheese and paprika to the sauce, stirring until the cheese is completely melted and glossy.
Fold the cooked pasta and broccoli into the cheese sauce until everything is evenly coated.
Cook it hands-free.
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