Melt 2 tablespoons unsalted butter in a saucepan over medium heat.

Add 2 tablespoons all-purpose flour and whisk continuously for about 1 minute, until it forms a smooth, bubbly paste (roux).

Gradually pour in 1 cup whole milk in a slow, steady stream while whisking constantly to prevent lumps.

Continue cooking and whisking over medium heat until the mixture thickens and coats the back of a spoon, about 2–4 minutes.

Reduce the heat to low. Add 1 1/2 cups shredded cheddar cheese a handful at a time, stirring until each addition is fully melted and the sauce is smooth.

Stir in 1/4 teaspoon salt and 1/4 teaspoon black pepper until evenly combined.

Place 2 cups steamed broccoli florets in a serving dish, pour the warm cheese sauce over the broccoli, and serve immediately.