Preheat the oven to 325 degrees F (165 degrees C).

Place broccoli in a steamer over 1 inch boiling water. Cover and cook until crisp-tender, 2 to 6 minutes. Drain.

Melt butter in a medium saucepan over medium heat. Add flour and stir, cooking until bubbly. Stir in onion, then milk. Bring to a boil and cook for 1 minute. Remove from heat and season with salt and pepper.

Add cheese and eggs and stir until cheese melts. Add broccoli and stir to coat. Transfer mixture to a 9x13-inch casserole dish.

Bake in the preheated oven until bubbly, about 30 minutes.