Start by preheating your oven to 350 degrees F and greasing your baking dish to prevent sticking. Bring a large pot of water to a boil, add the broccoli florets, and cook them for just 3 to 4 minutes. Drain them well and set them aside, as they will finish cooking in the oven later.
Melt the butter in a saucepan over medium heat, then whisk in the flour, garlic powder, and paprika. Cook this mixture for a minute or two until it smells nutty and looks golden, which removes the raw flour taste. This base is crucial for thickening your sauce properly.
Slowly pour in the half & half while whisking constantly to ensure no lumps form in your sauce. Let it simmer until it thickens, then remove from heat and stir in the shredded cheese, salt, and pepper until smooth. The residual heat will melt the cheese perfectly without separating it.
Pour your luscious cheese sauce directly over the drained broccoli florets and toss gently to coat every piece. Transfer the mixture into your prepared baking dish, spreading it out into an even layer. Make sure the cheese sauce is distributed well so every bite is flavorful.
In a small bowl, crush the butter crackers into coarse crumbs using your hands or a rolling pin. Mix these crumbs with the melted butter and any remaining shredded cheese you set aside. This combination creates the signature crispy texture that everyone loves.
Sprinkle the cracker mixture evenly over the top of the broccoli in the casserole dish. Bake uncovered at 350 degrees F for about 20 to 25 minutes until the sauce is bubbly and the top is golden brown. If you want it browner, broil it for a minute at the end.
Remove the casserole from the oven and let it stand for about 10 minutes before serving. This resting period allows the sauce to thicken slightly and makes it much easier to scoop. Serve it warm and enjoy the creamy, cheesy goodness.