Preheat oven to 325 degrees F (165 degrees C).
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.
Bake in preheated oven for 30 minutes.
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