Place broccoli on a cutting board and use a sharp knife to remove the tough part of the stems (usually the lower 3 to 4 inches). Cut broccoli into 4 even portions.
Bring a large pot of water to a boil with 1 teaspoon salt. Add broccoli and cook until tender, 3 to 6 minutes. Drain broccoli in a colander set in sink. Using tongs, transfer to a 2-quart casserole or serving dish.
In a heavy 4-quart saucepan, melt butter over medium heat. Add flour, remaining 1 teaspoon salt, and pepper and cook, stirring constantly, 2 minutes. Whisk in milk and bring to a boil, whisking occasionally. Reduce heat to medium-low and simmer, stirring occasionally, 2 minutes.
Remove from heat and add cheese. Stir well. Pour sauce over broccoli and serve.