Place broccoli in a steamer over 1 inch of boiling water, and cover. Cook until tender but still firm, about 2 to 6 minutes.
Drain.
Melt butter in a medium saucepan over medium heat and add flour; cook until bubbly. Stir in onion and gradually add milk, stirring well. Bring to a boil and cook for 1 minute.
Remove from heat and season with salt and pepper. Stir in cheese and eggs; mix well.
Combine mixture with broccoli and transfer to a 9 x 13 inch casserole dish.
