Preheat oven to 350°F (175°C) and lightly grease a 2-quart baking dish.
Steam the broccoli florets for 3-5 minutes until tender-crisp; drain thoroughly to prevent a watery casserole.
In a medium saucepan over medium heat, melt 1/4 cup butter.
Whisk in the flour and cook for 1 minute to form a roux, ensuring it does not brown.
Gradually whisk in the milk and simmer until the mixture thickens and coats the back of a spoon.
Remove the sauce from heat and stir in 1.5 cups of shredded cheddar cheese until smooth; season with salt and pepper.
Arrange the steamed broccoli in the prepared baking dish and pour the cheese sauce evenly over the florets.
In a small bowl, combine the crushed crackers, melted butter, and the remaining 0.5 cup of cheddar cheese.
Sprinkle the cracker mixture over the casserole.
Bake for 25-30 minutes until the sauce is bubbling and the topping is golden brown.